UC-NRLF 


GIFT   OF 
MICHAEL  REESE 


THE 


RAILROAD    SPIRAL. 

THE    THEORY    OF    THE 

COMPOUND  TRANSITION   CURVE 


REDUCED   TO 


PRACTICAL   FORMUL/E    AND    RULES    FOR 
APPLICATION    IN   FIELD   WORK; 


COMPLETE    TABLES    OF    DEFLECTIONS    AND     ORDINATES 
FOR    FIVE    HUNDRED    SPIRALS. 


KY  • 

WILLIAM    H.    SEARLES,   C.E., 

MEMBER    AMERICAN    SOCIETY    OF   CIVIL   KNG1NKERS, 
AUTHOR   "FIELD    ENGINEERING." 


UNIVERSITY! 


NEW   YORK  : 

JOHN   WILEY  &   SONS. 
1882. 


COPYRIGHT, 

1882, 
BY  JOHN  WILEY  &  SONS. 


PRE  FACE. 


THE  object  of  this  work  is  to  reduce  the  well-known 
theory  of  the  cubic  parabola  or  multiform  compound 
curve,  used  as  a  transition  curve,  to  a  practical  and  con- 
venient form  for  ordinary  field  work. 

The  applicability  of  this  curve  to  the  purpose  in- 
tended has  been  fully  demonstrated  in  theory  and  prac- 
tice by  others,  but  the  method  of  locating  the  curve  on 
the  ground  has  been  left  too  much  in  the  mazes  of 
algebra,  or  else  has  been  described  as  a  system  of  off- 
sets, or  fudging.  Where  a  system  of  deflection  angles 
has  been  given,- the  range  of  spirals  furnished  has  been 
much  too  limited  for  ge^r&^nictice.  In  consequence 
the  great  majority  of  engineers  have  contented  them- 
selves with  locating  circular  curves  only,  leaving  to  the 
trackman  the  task  of  adjusting  the  track,  not  to  the 
centres  given  near  the  tangent  points,  but  to  such  an 
approximation  to  the  spiral  as  he  could  give  "by  eye." 

The  method  here  described  is  that  of  transit  and 
chain,  analogous  to  the  method  of  running  circular 
curves  ;  it  is  quite  as  simple  in  practice,  and  as  accu- 
rate in  result.  No  offsets  need  be  measured,  and  the 
curve  thus  staked  out  is  willingly  followed  by  the  track- 
men because  it  "  looks  right,"  and  is  right. 

The  preliminary  labor  of  selecting  a  proper  spiral  for 
a  given  case,  and  of  calculating  the  necessary  distances 
to  locate  it  at  the  proper  place  on  the  line,  is  here  ex- 
plained, and  reduced  to  the  simplest  method.  Many  of 

iii 


IV  PREFACE. 

the  quantities  required  have  been  worked  out  and  tabu- 
lated once  for  all,  leaving  only  those  values  to  be  found 
which  are  peculiar  to  the  individual  case  in  hand.  A 
large  number  of  spirals  are  thus  prepared,  and  their 
essential  parts  are  given  in  Table  III. 

In  section  22  is  developed  the  method  of  applying 
spirals  to  existing  circular  curves,  without  altering  the 
length  of  line,  or  throwing  the  track  off  of  the  road  bed, 
an  important  item  to  roads  already  completed.  Table 
V.  contains  samples  of  this  kind  of  work  arranged  in 
order,  so  that,  by  a  simple  interpolation,  the  proper  se- 
lection can  be  made  in  a  given  case. 

The  series  of  spirals  given  in  Table  III.  are  obtained 
by  a  simple  variation  of  the  chord-length,  while  the  de- 
flections and  central  angles  remain  constant.  This  is 
the  converse  of  our  series  of  circular  curves,  in  which 
the  chord  is  constantly  100  feet,  while  the  deflections 
and  central  angles  take  a  series  of  values. 

The  multiform  compound  curve  has  been  chosen  as 
the  basis  of  the  system,  rather  than  the  cubic  parabola, 
because,  while  there  is  no  practical  difference  in  the 
two,  the  former  is  more  in  keeping  with  ordinary  field 
methods,  and  is  far  more  convenient  for  the  calculation 
and  tabulation  of  values  in  terms  of  the  chord-unit,  or  of 
measurement  along  the  curve.  While  the  several  com- 
ponent arcs  of  the  spiral  are  thus  assumed  to  be  circu- 
lar, yet  the  chord-points  are  points  of  a  true  spiral,  to 
which  the  track  naturally  conforms  when  laid  according 
to  the  chord-points  given  as  centres. 

The  "  Railroad  Spiral "  is  in  the  nature  of  a  sequel  to 
"  Field  Engineering  ;  "  the  same  system  of  notation  is 
adopted,  and  any  tables  referred  to,  but  not  given  here, 
will  be  found  in  that  work. 

WM.  H.  SEARLES,  C.  E. 

NEW  YORK,  July  i,  1882. 


CONTENTS. 


CHAPTER    I. 

INTRODUCTION. 
SECTION  PAGE 

1.  Objections  to  simple  circular  curves I 

2.  Office  of  the  spiral 2 


CHAPTER   II. 

THEORY    OF   THE    SPIRAL. 

3.  Description  of  the  spiral 3 

4.  Co-ordinates  of  the  spiral 3 

5.  Deflection  angles  from  the  main  tangent . . . .  5 

6.  Deflection  angles  from  an  auxiliary  tangent 6 

7.  The  chord-length  as  a  variable g 

Construction  of  Table  of  Co-ordinates 10 

9.   Elements  of  the  spiral 10 

10.  Selection  of  a  spiral 1 1 

CHAPTER    III. 

ELEMENTARY    PROBLEMS. 

11.  To  find  a  long  chord  SL 13 

12.  To  find  the  tangents  SE  and  EL 13 

13.  To  find  a  long  chord  QL.  „ 14 

14.  To  find  the  tangents  QE'  and  EL 15 

15.  To  find  the  tangent-distance  Ts  —  SV 16 

16.  To  find  7's  approximately 17 

17.  To  find  the  radius  R1  in  terms  of  Ts  and  spiral 17 

V 


VI  CONTENTS. 

SECTION  PAGE 

18.  To  find  diff.  R'  in  terms  of  d'iff.  Ts 19 

19.  To  find  the  external  distance  Es 20 

20.  To  find  the  radius  R'  in  terms  of  Es  and  spiral 21 

21.  To  find  diff.  R'  in  terms  of  diff.  x  for  Es  constant 23 

CHAPTER    IV. 

SPECIAL     PROBLEMS. 

22.  Given,   a  simple    curve,  to    replace    it   by  another  with 

spirals  ;  length  of  line  unchanged 25 

a.  To  find  the  radius  R' 26 

b.  To  find  the  offset  h 26 

c.  To  find  the  distance  d  —  A  S. 27 

d.  To  find  lengths  of  old  and  new  lines 27 

e.  To  select  a  suitable  spiral 28 

f.  To  find  diff.  h  in  terms  of  diff.  R' 29 

23.  Given,  a  simple  curve,  to  apply  spirals  without  change  of 

radius 32 

24.  Given,  a  simple  curve,  to  compound  it  for  spirals  without 

disturbing  the  middle  portion 34 

25.  Given,  a  compound  curve,  to  replace  it  by  another,  with 

spirals  ;  length  of  line  unchanged 36 

26.  Given,  a  compound  curve,  to  apply  spirals  without  change 

of  radii 40 

27.  Given,  a  compound  curve,  to  introduce  spirals  without  dis- 

turbing the  P.  C.  C 42 

CHAPTER  V. 

FIELD   WORK. 

28.  To  locate  a  spiral  from  S  to  L 45 

29.  To  locate  a  spiral  from  L  to  S 46 

30.  To  interpolate  the  regular  stations 47 

31.  Choice  of  method  for  locating  spirals    47 

32.  To  locate  a  spiral  by  ordinates 48 

33.  Use  of  spirals  on  location  work 48 

34.  Description  of  line  with  spirals 48 

35.  Elevation  of  outer  rail  on  spirals 49 

36.  Monuments 49 

37.  Keeping  field-notes 49 


CONTENTS. 


TABLES. 

PAGE 

I.  Elements  .of  the  spiral  of  chord-length  TOO 50 

II.  Deflection  angles  for  the  spiral 52 

II.  Co-ordinates  and  Degree  of  curve  of  the  spiral 58 

IV.  Functions  of  the  spiral  angle  s 77 

V.  Selected  spirals  for  unchanged  length  of  line.     §  22 78 


UNIVERSITY; 

A 

^\^      Sr     it  C)Jd'  ^*t    V*w»      '/ 

THE   RAILROAD   SPIRAL. 


CHAPTER   I. 

INTRODUCTION. 

I.  ON  a  straight  line  a  railway  track  should  be  level 
transversely  ;  on  a  curve  the  outer  rail  should  be  raised 
an  amount  proportional  to  the  degree  of  curve.  At  the 
tangent  point  of  a  circular  curve  both  of  these  condi- 
tions cannot  be  realized,  and  some  compromise  is  usually 
adopted,  by  which  the  rail  is  gradually  elevated  for 
some  distance  on  the  tangent,  so  as  to  gain  at  the  tan- 
gent point  either  the  full  elevation  required  for  the 
curve,  or  else  three-quarters  or  a  half  of  it,  as  the  case 
may  be.  The  consequence  of  this,  and  of  the  abrupt 
change  of  direction  at  the  point  of  curve,  is  to  give  the 
car  a  sudden  shock  and  unsteadiness  of  motion,  as  it 
passes  from  the  tangent  to  the  curve. 

The  railroad  spiral  obviates  these  difficulties  entirely, 
since  it  not  only  blends  insensibly  with  the  tangent  on 
the  one  side,  and  with  the  circle  on  the  other,  but  also 
affords  sufficient  space  between  the  two  for  the  proper 
elevation  of  the  outer  rail.  Moreover,  since  the  curva- 
ture of  the  spiral  increases  regularly  from  the  tangent 
to  the  circle,  and  the  elevation  of  the  outer  rail  does 
the  same,  the  one  is  everywhere  exactly  proportional  to 
the  other,  as  it  should  be.  The  use  of  the  spiral  allows 

i 


2  THE    RAILROAD    SPIRAL. 

the  track  to  remain  level  transversely  for  the  whole 
length  of  the  tangent,  and  yet  to  fie  fully  inclined  for 
the  whole  length  of  the  circle,  since  the  entire  change 
in  inclination  takes  place  on  the  spiral. 

2.  The  office  of  the  spiral  is  not  to  supersede  the  cir- 
cular curve,  but  to  afford  an  easy  and  gradual  transition 
from  tangent  to  curve,  or  vice  versa,  in  regard  both  to 
alignment  and  to  the  elevation  of  the  outer  rail.  A 
spiral  should  not  be  so  short  as  to  cause  too  abrupt  a 
rise  in  the  outer  rail,  nor  yet  so  long  as  to  render  the 
rise  almost  imperceptible,  and  therefore  difficult  of  ac- 
tual adjustment.  Within  these  limits  a  spiral  may  be 
of  any  length  suited  to  the  requirements  of  the  curve 
or  the  conditions  of  the  locality.  To  suit  every  case  in 
practice  an  extensive  list  of  spirals  is  required  from 
which  to  select. 


THEORY    OF    THE    SPIRAL. 


CHAPTER   II. 

THEORY    OF    THE    SPIRAL. 

3.  THE  Railroad  Spiral  is  a  compound  curve  closely 
resembling  the  cubic  parabola  ;    it  is  very  flat  near  the 
tangent,  but  rapidly  gains  any  desired  degree  of  curva- 
ture. 

The  spiral  is  constructed  upon  a  series  of  chords  of 
equal  length,  and  the  curve  is  compounded  at  the  end 
of  each  chord.  The  chords  subtend  circular  arcs,  and 
the  degree  of  curve  of  the  first  arc  is  made  the  com- 
mon difference  for  the  degrees  of  curve  of  the  suc- 
ceeding arcs.  Thus,  if  the  degree  of  curve  of  the  first 
arc  be  o°  ic/j  that  of  the  second  will  be  o°  20',  of  the 
third,  o°  30',  &c. 

The  spiral  is  assumed  to  leave  the  tangent  at  the  be- 
ginning of  the  first  chord,  at  a  tangent  point  known  as 
the  Point  of  Spiral,  and  designated  by  the  initials  P.  S.r 
or  on  the  diagrams  by  the  letter  S. 

4.  To  determine  the  co-ordinates  of  the  sev- 
eral chord  extremities,  let  the  point  S  be  taken  as 
the  origin  of  co-ordinates,  the  tangent  through  S  as  the 
axis  of  Y,  and  a  perpendicular  through  S  as  the  axis  of 
X.     Then  x,  y,  will  represent  the  co-ordinates  of  any 
point  of  compound  curvature  in  the  spiral,  x  being  the 
perpendicular  offset  from  the  point  to  the  tangent,  and 
y  the  distance  on  the  tangent  from  the  origin  to  that 
offset. 

For  the  purpose  of  calculation  let  us  assume  100  feet 
as  the  chord-length,  and  o°  10'  as  the  degree  of  curve  of 


THE    RAILROAD    SPIRAL. 


the  first  arc  of  a  given  spiral.  Then,  since  the  degree 
of  curve  is  an  angle  at  the  centre  of  a  circle  subtended 
by  a  chord  of  100  feet,  the  central  angle  of  the  first 
chord  is  10',  of  the  second  20',  of  the  third  30',  &c.,  and 
the  angles  which  the  chords  make  with  the  tangent  are  : 

For  ist  chord,  Y    x   10'  =     5' 

"    2d    -    u    10'    +     1  X  20  =  20' 

"    3d        "    10'   +  20'  4-  i  x  30'           =  45' 

"  4th        "     10'  +  20    +  30  +  i  x  40   =  80' 

&c.,                              &c.,  &c., 

or  in  general  the  inclination  of  any  chord  to  the  tan- 
gent at  S  is  equal  to  half  the  central  angle  subtended 
by  that  chord  added  to  the  central  angles  of  all  the 
preceding  chords.  If  now  we  consider  the  tangent  as 
a  meridian,  the  latitude  of  a  chord  will  be  the  product  of 
the  chord  by  the  cosine  of  its  inclination,  and  its  depart- 
ure will  be  the  product  of  the  chord  by  the  sine  of  its 
inclination  to  the  tangent.  A  summation  of  the  several 
latitudes  for  a  series  of  chords  will  give  us  the  required 
values  of  _>',  and  a  summation  of  the  several  departures 
will  give  us  the  required  values  of  x.  By  the  aid  of  a 
table  of  sines  and  cosines,  we  may  therefore  readily  pre- 
pare the  following  statement  : 


Chord. 

Inclin. 
to  tang. 

Dep.  — 
100  sine. 

x. 

Lat.  •= 
ioo  cosine. 

y- 

I 

o°  05' 

0.145 

•145 

100.000 

TOO.OOO 

2 

0°   20' 

0.582 

.727 

99.998 

199.998 

3 

o°  45  ' 

1.309 

2.036 

99.991 

299.989 

4 

1°   20' 

2.327 

4-363 

99.979 

399.968 

&c. 

&c.. 

&c. 

In  this  manner  Table  I.  has  been  constructed. 


THEORY    OF    THE    SPIRAL. 


5.  To  calculate  the  deflection  angles  of  the 
Spiral ;  Inst.  at  S.  If  in  the  diagram,  Fig.  i,  we 
draw  the  long  chords  82,  83,  84,  &c.,  g 
we  may  easily  determine  the  angle  /, 
which  any  long  chord  makes  with  the 
tangent  by  means  of  the  co-ordinates 
of  the  further  extremity  of  the  chord, 
for 

x 

tan  /  =  — . 

y 

Having  calculated  a  series  of  values 
of  the  angle  /,  we  may  lay  out  the 
spiral  on  the  ground  by  transit  deflec- 
tions from  the  tangent,  the  transit  t>£ 
ing  at  the  point  S. 

The  statement  of  the  calculation  is 
as  follows  :  FIG 


Point. 

X 

/ 

tan  /  =  -  . 

i 

y 

I 

.145 

too.ooo 

.00145 

o°  05'  oo" 

2 

.727 

199.998 

.00364 

12'  30" 

3 

2.036 

299.989 

.00679 

23'  20" 

4 

4-363 

399.968 

.01091 

37'  30" 

&c. 

&c. 

The  values  of  /  are  more  readily  found  by  logarithms 
however,  since 

log  tan  /  =  log  x  —  logy. 
By  this  formula  the  first  part  of  Table  II.  (Inst.  at  S) 


THE    RAILROAD    SPIRAL. 


FIG.  2. 


has  been  calculated,  and  these  are 
the  only  deflections  needed  for  field 
use  when  the  entire  spiral  is  visible 
from  S. 

6.  To  calculate  the  deflection 
angles  when  the  transit  is  at  any 
other  chord-point  than  S  :  Sup- 
pose the  transit  at  point  I,  Fig.  2. 

In  the  diagram  draw  through  the 
point  i  a  line  parallel  to  the  tangent 
at  S,  and  also  the  long  chords  1-3, 
1-4,  &c.,  and  let  a{  represent  the 
angle  between  any  one  of  these  long 
chords  and  the  parallel.  Then,  from 
the  right-angled  triangles  of  the  dia- 
gram we  have  the  following  expres- 
sions : 


For  point  2,  tan  #,  =  -       — ^  =  — — —Q  =  .00582. 

y*  —y\      99-998 

xz—  .\\         1.891 

"     3,  tan  a,  =  -  -~~  =  .00945. 

y,  -yi        I99*989 


4,  tan  #!  = 
&c., 


4.218 

299.968 

&c. 


=  .01411. 


But  these  are  better  worked  by  logarithms,  and  the 
values  of  al  found  directly  from  the  logarithmic  tan- 
gent. 

Let  s  —  the  spiral  angle  =  the  angle  subtended  by 
any  number  of  spiral  chords,  beginning  at  S.  Then 
s  =  the  sum  of  the  central  angles  of  the  several  chords 
considered  ;  and  it  therefore  equals  the  angle  between 


0       OO 

0 

10' 

10' 

20' 

3°; 

3°' 

1°    00- 

40' 

o           t 
I       40 

&c. 

THEORY    OF    THE   SPIRAL.  7 

the  tangent  at  S  and  a  tangent  at  the  last  point  consid- 
ered.    The  series  of  values  of  the  angle  s  is  as  follows : 

Point.         Angle  under  single  chord.    Angle  f. 

S 

I 

2 

3 

4 
&c. 

Since  the  values  of  a\  found  above  are  deflections 
at  point  i  from  a  parallel  to  the  main  tangent,  it  is  evi- 
dent that  if  we  subtract  from  each  the  value  of  s  for 
point  i,  or  10',  we  shall  have  the  deflections,  /,  from  an 
auxiliary  tangent  through  the  point  i,  which  we  require 
for  use  in  the  field.  The  statement  is  as  follows  : 

Instrument  at  point  i  ;  (s  =  10'). 

Point.  Angle «,.  Angle/. 

2  20"  I0f 

3  32'  30"  22'  30" 

4  48'  20"  38'  20" 
&c.,             &c.,  &c. 

The  instrument  will  read  zero  on  the  auxiliary  tan- 
gent through  point  i  where  it  stands,  and  of  course  the 
back  deflection  over  the  circular  arc  Si  is  05'.  Hence 
we  have  the  complete  table  of  deflections  when  the 
instrument  is  at  point  i. 

Similarly,  if  we  suppose  the  instrument  to  be  at 
point  2,  we  shall  have  the  statement : 

Point.  _ 

3  tan  a>2  =  — =  =  .01018.  - 

y*  —_f*         99-991 

4  tan  a*  —  — —  -3^—3 —  _  OIr-27. 

y±  —y*      199-97° 

&c.,  &c., 


8 


THE    RAILROAD    SPIRAL. 


and  since  for  point  2,  s  =  20',  we  have  : 

Point.  Angle  a^.  Angle  i. 

3  °°  55'  '  .  o°  15' 


30 


&c., 


32   30 
&c. 


The  instrument  will  read  zero  on  the  auxiliary  tangent 
through  the  point  2,  the  back  deflection  to  the  point  i 
is  half  the  central  angle  under  the  second  chord,  or  10', 
and  the  back  deflection  to  S  is  the  difference  between 
S?  and  the  deflection  at  S  for  point  2,  or  30'  —  12'  30"  — 
17'  30".  We  thus  may  complete  the  table  of  deflections 
for  the  instrument  at  point  2. 

By  a  similar  process  the  deflections  required  at  any 
other  chord-point  may  be  deduced.  It  should  be  noted, 
however,  in  forming  the  table,  that  the  back  deflection 

5  \ ,y    to  any  point  is  equal  to  the  value 

of  s  for  the  place  of  the  instru- 
ment, less  the  value  of  s  for  the 
back-point,  less  the  forward  de- 
flection at  the  back-point  for  the 
place  of  the  instrument.  This  is 
obvious  from  an  inspection  of  the 
triangle  formed  by  the  two  auxil- 
iary tangents  and  the  chord  join- 
ing the  two  points  in  question. 

Thus,  Fig.  3,  when  the  instru- 
ment is  at  point  4,  the  back  de- 
flection for  point  2  is  equal  to 
100'  —  30'  —  32'  30"  =  37'  30." 

In  the  manner  above  described 
has  been  calculated  the  complete 
table  of  deflections  from  auxiliary 
tangents  at  chord-points,  for  every  chord-point  of  the 
spiral  up  to  point  20,  Table  II.  It  is  evident,  that  by 


FIG. 


THEORY    OF    THE    SPIRAL.  9 

means  of  this  table  the  entire  spiral  may  be  located,  the 
transit  being  set  over  any  chord-point  desired,  while  the 
chain  is  carried  around  the  curve  in  the  usual  manner ; 
also,  that  the  curve  may  be  laid  out  in  the  reverse  direc- 
tion from  any  chord-point  not  above  the  2oth,  since  all 
the  back  deflections  are  also  given. 

7.  Variation  in  the  chord-length. 

We  have  thus  far  assumed  the  spiral  to  be  constructed 
upon  chords  of  100  feet,  but  it  is  evident  that  such  a 
spiral  would  be  entirely  too  long  for  practical  use  ;  it 
would  be  1700  feet  long  before  reaching  a  3°  curve. 

We  must,  therefore,  assume  a  shorter  chord ;  but  in 
so  doing  it  will  not  be  necessary  to  recalculate  the 
angles  and  deflections,  for  these  remain  the  same  whatever 
be  the  chord-length.  By  shortening  the  chord-length  we 
merely  construct  the  spiral  on  a  smaller  scale.  The 
values  of  x  and  y  and  of  the  radii  of  the  arcs  at  corre- 
sponding points  are  proportional  to  the  chord-lengths, 
and  the  degrees  of  curve  for  corresponding  chords  are 
(nearly)  inversely  proportional  to  the  same. 

Thus  for  any  chord-length  c  we  have  : 

c 
x  :  x1QO  ::  c  :  100,     or     x  =  —  ^100- 

100 

c 

y  :  Vioo  ::  c  :  100,     or     y  —  —  y100. 

100 

c 

Rs  :  jRiM  ::  c  :  100,     or  R3  —  — ^?100. 

100 

Let  Ds  =  the  degree  of  curve  due  to  radius  JRS  ,  and 
.Z}100  =  the  degree  of  curve  due  to  radius  ^10o?  then, 

100  100 

A«  ~  : — TTv  and  -*MOO  — 


2  sin 
whence 

IOO 

sin  1  D,  =  —  sin  4-  Z>JOo, 

T*  C 


TO  THE    RAILROAD    SPIRAL. 

in  which  Ds  is  the  degree  of  curve  upon  any  chord  in  a 
spiral  of  chord-length  c,  and  Z>100  is  the  degree  of  curve 
upon  the  corresponding  chord  in  the  spiral  of  chord- 
length  100. 

Accordingly,  if  we  assume  a  chord-length  of  10  feet 

the  values  of  x  and  y  will  be  • —  of  those  calculated  for 

100 

a  chord-length  of  100  feet,  while  the  degree  of  curve 
on  each  chord  will  be  (nearly)  10  times  as  great  as  be- 
fore. 

8.  In  the  construction  of  Table  III.,  we  have  as- 
sumed the  chord  to  have  every  length  successively  from 
10  feet  to  50  feet,  varying  by  a  single  foot,  and  have 
calculated   the  corresponding  values  of  x,  y  and  Z)8 . 
The    logarithm   of  x  is  also  added,  and  the,  length  of 
spiral  nc. 

We  are  thus  furnished  with  41  distinct  spirals,  but 
since  the  same  spiral  may  be  taken  with  a  different 
number  of  chords  (not  less  than  three)  to  suit  different 
cases,  the  variations  which  the  tables  furnish  amount  to 
no  less  than  500  spirals,  some  one  or  more  of  which 
will  be  adapted  to  any  case  that  can  arise.  The  maxi- 
mum length  of  spiral  has  been  taken  at  400  feet  ;  the 
shortest  spiral  given  is  3x10  feet  =  30  feet.  Be- 
tween these  limits  may  be  found  spirals  of  various 
lengths. 

9.  The  elements  of  a  spiral  are  : 

D^  The  degree  of  curve  on  the  last  chord, 
«,  The  number  of  chords  used, 
cy  The  chord-length, 
n  x  cy  The  length  of  spiral, 

s,  The  central  angle  of  the  spiral, 
xy  y,  The  coordinates  of  the  terminal  point. 
Every  spiral  must  terminate,  or  join  the  circular  curve 


THEORY    OF    THE    SPIRAL.  II 

at  a  regular  chord-point  of  which  the  coordinates  are 
known. 
10.  To  select  a  spiral. 

The  terminal  chord  of  a  spiral  must  subtend  a  degree 
of  curve  less  than  that  of  the  circular  curve  which  fol- 
lows, but  the  next  chord  beyond  (were  the  spiral  pro- 
duced) must  subtend  a  degree  of  curve  equal  to  or 
differing  but  a  little  from  that  of  the  circular  curve. 

Thus,  if  the  circle  were  a  10  degree  curve,  the  spiral 
may  consist  of  5  chords  10  feet  long  (the  degree  of 
curve  on  the  6th  chord  being  10°  oo'  45"),  or  of  15 
chords  26  feet  long  (the  degree  of  curve  on  the  i6th 
chord  being  10°  16'  09"),  the  length  of  'spiral  is  50  feet 
in  one  case  and  390  in  the  other ;  between  these  limits 
the  tables  furnish  15  other  spirals  of  intermediate  length, 
all  adapted  to  join  a  10  degree  curve. 

We  may  therefore  introduce  one  more  condition  which 
will  fix  definitely  the  proper  spiral  to  employ.  If  the 
length  of  spiral  be  assumed,  we  seek  in  the  tables  those 
values  of  n  and  c  which  are  consistent  with  the  required 
value  of  Ds  for  (;/  -h  i),  at  the  same  time  that  their 
product,  nc,  equals  as  nearly  as  may  be  the  assumed 
length  of  spiral.  Thus,  if  with  a  10  degree  curve  a 
length  of  about  130  feet  were  desirable,  we  should  select 
either 

n  =  8,  f=  15,  D9  —  10°  oo'  45";     nc  —  120  ft; 
or  ;z  =-9,  c=  16,  Ds  —  10°  25'  51";     nc  —  144  ft. 

Ds  is  always  taken  for  (n.+  i).  When  circumstances 
permit,  a  chord-length  of  about  30  feet  will  give  the 
best  proportioned  spirals.  With  a  30  foot  chord-length 
the  length  of  spiral  will  be  about  770  times  the  super- 
elevation of  the  outer  rail  at  a  velocity  of  35  miles  per 
hour. 


12  THE    RAILROAD    SPIRAL. 

The  value  of  s  depends  on  the  number  of  chords  (n) 
and  is  independent  of  the  chord-length.  If  the  angle  s 
were  selected  from  the  table,  this  would  fix  the  number 
n,  and  we  must  then  choose  the  chord-length  c  so  as  to 
give  the  proper  value  of  Ds.  Thus,  if  s  were  assumed 
=  9°  10'  then  n  =  10,  and  €••=•  18  ft.  or  19  ft.,  giving 
Ds  —  10°  iir  54"  or  9°  39'  36"  to  suit  a  10  degree  curve, 
and  making  the  length  (nc)  of  the  spiral  either  170  or 
1 80  ft.,  according  to  the  spiral  selected. 

The  coordinates  (<x,y)  depend  on  the  values  of  both 
n  and  c.  They  are  used  in  solving  the  problems  of 
the  spiral,  being  taken  directly  from  Table  III.  for  this 
purpose,  under  the  value  of  c  and  opposite  the  value 
of  n. 


ELEMENTARY    PROBLEMS. 


CHAPTER  III. 

ELEMENTARY    PROBLEMS. 

ii.  To  find  the  length  C  of  any  long  chord 
beginning  at  the  point  of  spiral  S.  Fig.  4.  Let 
L  be  the  other  extremity  of  the  long 
chord,  x,  y  the  coordinates  of  L,  and 
/  the  deflection  angle  YSL  at  S  for 
the  point  L. 


Then 


or 


C  =  - 
cos  i 


.  (I.) 


——. 

sin  / " 


The  values  of  x,  y  and  /  are  found 
in  Tables  III.  and  II. 

Example.  In  the  spiral  of  chord- 
length  =  30  ft.  what  is  the  length  of 
the  long  chord  from  S  to  the  loth 
point  ? 


FIG.  4: 


From  Table  III., 


log  x  1.224491 

/     3°  12'  28"  log  sin  8.747853 


C        299.66     Ans. 


2.476638 


12.  To  find  the  lengths  of  the  tangents  from 
the  points   S  and    L  to    their  intersection  E. 

Fig.  4.     Let  x,  y  be  the  coordinates  of  L,  and  s  the 


14  THE    RAILROAD    SPIRAL. 

spiral  angle  for  the  point  L.     Then  s  —  the  deflection 
angle  between  the  tangents  at  E,  and 


LE  =  - —  SE^y  —  x  cot  s  .     .     .     .      (2.) 

sm  s 


The  values  of  xy  y  and  s  are  found  in  Tables  III.  and 
IV. 

Example.  In  the  spiral  of  chord-length  40  extending 
to  the  pth  point,  what  are  the  tangents  LE  and  SE  ? 

From  Table  III.,  log  x          1.219075 

"      IV.,  s  7°  30'  log  sin       9.115698 

LE  =  126.87  2.103377 

log  A*  1.219075 

s  7°  30'  log  cot       0.880571 

125.790  2.099646 

y      359-352 


SE  =  233.562 

13.  To  find  the  length  C  of  any  long  chord 

KL.  Fig.  4.  Let  x,  y  be  the  coordinates  of  L,  and 
xr,  y  the  coordinates  of  K  ;  and  let  a  be  the  angle  LKN 
which  LK  makes  with  the  main  tangent,  and  /  the  de- 
flection angle  KLE',  and  /"  the  deflection  angle  LKE'. 
Then  a  =  (s  —  i)  at  the  point  L,  =  (s!  +  /')  at  K. 


KN 

KL  =  -  T-^r-T          or 
cos  LKN 


(3.) 


cos  a 
Example.  In  the  spiral  of  chord-length   18  what  is  the 


ELEMENTARY    PROBLEMS.  15 


length  of  the  long  chord  from  point  12  to  point  20? 
Here  K  =  12  and  L  =  20  =  n. 


From  Table  III.,      y  346.47§TJ  fl IV  \ 
y   214.?  ' 


log         2.119352 
From  Table  II.,    /     13° 

f       I0°-07'23" 


.  * .  a     23°  07'  23"  log  cos   9.963629 

.'.     C—  143-13  2-I55723 

14.  To  find  the  lengths  of  the  tangents  from 
any  two  points  L  and  K  to  their  intersection  at 

E'.  Fig.  4.  Let  s,  sf  be  the  spiral  angles  for  the  points 
L  and  K  respectively.  Then  (s  —  /)  —  the  deflection 
angle  between  tangents  at  E'.  Having  first  found  C  — 
LK  by  the  last  problem  we  have  in  the  triangle  LKE' 

TTT'  c  sin  *'  W-       ^sm/'  (A  \ 

L.&      =     — -. jr  KJl.     —    — 7 - — 7T    •     •     (4-) 

sin  (s  —  s)  sin  (s  —  s  ) 

Example.  In  the  spiral  of  chord-length  18  what  are 
the  tangents  for  the  points  12  and  20  ? 

By  last  example,  C  log        2.155723 

From  Table  IV., 

(s  -  s)  35°  n  13°  =  22°  log  sin  9-573575 

2.582148 
From  Table  II.,      i      10°  07'  23"  log  sin  9.244927 


.',   LE'  =  67.15  1.827075 

Again:  2.582148 

Table  II. ,                 /  11°  52'  37''  log  sin  9.313468 

,-.  KE'  =  78.635  1.895616 


16 


THE    RAILROAD    SPIRAL. 


FlG 


15.  Given  :  A  circular  curve 
and  spirals  joining  two  tangents, 
to  find  the  tangent  dis- 
tance T,  =  VS.  Fig.  5. 

Let  S  be  the  point  of  spiral, 
V  the  intersection  of  the  tan- 
gents, SL  the  spiral,  LH  one 
half  the  circular  curve,  and  O 
its  centre.  In  the  diagram 
draw  GLI  parallel  to  the  tan- 
Is,  gent  VS,  and  GN,  LM,  and 
OI  perpendicular  to  VS.  Join 
OL  and  OV. 


Then 
IOL  =  s  ;  IOV  =  i  A  ;  OL  =  R'  ;  SM  =  y  ;  LM  =  x. 

Now  SV  =  SM  4-  NV  -f  MN. 

But  NV  =  GN  .  tan  VGN  =  x  tan  \  A. 

MN  =  GL  =  OL^££  =  R'  Sin  (i  A  ~  S]- 


Hence 


sin  OGI  cos  ^A 

,  sin  (|  A  —  s) 


•  ...(s.) 


Example.  Let  the  degree  of  the  circular  curve  be 
£>'  =  7°  20',  and  the  angle  between  tangents,  A  =42°. 
Let  the  spiral  values  be  c  =  2.3  ;  n  —  9  .  ' .  s  =  7°,3o'« 
Then  by  the  last  equation  and  the  tables, 


y 

x 


21 


206.627 


log 
log  tan 


0.978743 
9-584i77 


3,6,55 


0.562920 


ELEMENTARY    PROBLEMS.  17 

R'  7°  20'  C  log  2.893118 

iA—  s   13°  30'  log  sin  9.368185 

•j-A  21°  a.  c.  log  cos  0.029848 

195.502  2.291151 


.  '  .  Ta  =  405-784 

l6.  When  an  approximate  value  of  T,  is  only  re- 
quired we  may  employ  a  more  convenient  formula 
derived  from  the  fact  that  the  line  OI  produced  bisects 
the  spiral  SL  very  nearly,  and  that  the  ordinate  to  the 
spiral  on  the  line  OI,  being  only  about  -g-  x,  may  be  neg- 
lected. Thus, 

Approx.          T,  —  If  tan  |A  +  i  nc.     .     .     (6.) 
Example.  Same  as  above. 

R'          7°  20'  C  log         2.893118 

I  A        21°  log  tan  9.5,84177-' 

300.1.  2.477295 

i»^  =  ix9X23       103.5 


.  '  .  71  ==  approx.  403.6      > 

Remark.  This  formula,  eq.  (6)  when  K  is"  taken  equal 
to  the  radius  corresponding  to  the  degree  of  curve 
Ds  for  (n  -f  i),  gives  practically  correct  results.  But 
as  in  practice,  the  value  of  R1  will  differ  somewhat  frcm 
the  radius  of  D#  so  the  value  of  T8  derived  from  this 
formula  will  differ  more  or  less  from  the  true  value,  as 
in  the  last  example. 

1/17.  Given  :  the  tangent  distance  Ty  —  SV,  and  the 
angle  A  ,  and  the  length  of  spiral  SL,  to  find  the  radius 
K  of  the  circular  curve,  LH,  Fig.  5.  The  length 


Ib-  THE    RAILROAD    SPIRAL. 

of  spiral  is  expressed  by  nc,  hence  we  have  from  the 
last  equation. 

approx.,  £'  =  (Ta  —  \nc)  cot  -JA.     .     .     .(7.) 

After  R'  is  thus  found,  the  values  of  n  and  c  are  to  be 
determined,  such  that,  while  their  product  equals  the 
given  length  of  spiral  as  nearly  as  may  be,  the  value  of 
Z>4  for  (n  -f  i)  shall  correspond  nearly  with  R '.  The 
values  of  n  and  c  are  quickly  found  by  reference  to 
Table  III. 

Examble.     Let  Ts  =  406,  A  =  42°,  and  nc  =  170. 

T9  —  \nc  321  log  2.5065 

lA          21°  log  COt.  0.4158 

Jt'  =•  say,  6°  51'  curve,  2.9223 

By  reference  to  Table  III.,  we  find  that  when  n  =  8 
and  c  =  22,  the  product  nc  being  176,  the  value  of  D3 
for  (n  +  i)  is  6°  49'  19",  and  this  is  the  best  spiral  to 
use  in  this  case.  But  as  this  spiral  is  longer  than  our 
assumed  one,  we  should  decrease  the  value  of  Jt'  some- 
what, if  we  would  nearly  preserve  the  given  value  of 
Ts.  For  instance,  assume  R'  —  radius  of  6°  54'  curve, 
and  using  the  same  spiral,  calculate  by  eq.  (4)  the  re- 
sulting value  of  TS9  and  we  shall  find  T»  =  408.646. 

As  this  is  an  exact  value  of  T*  for  the  values  of  R\  n 
and  c  last  assumed,  and  is  also  a  close  approximation  to 
the  value  first  given,  it  will  probably  answer  the  purpose 
completely.  If,  however,  for  any  *  reason  the  precise 
value  of  Ts  —  406  is  required,  we  may  find  the  precise 
radius  which  will  give  it  by  the  following  problem. 

l8.  Given:  a  curve,  and  spiral,  and  tangent-distance, 


ELEMENTARY    PROBLEMS.  19 

T»  to  find  the  difference  in  £'  corresponding  to 
any  small  difference  in  the  value  of  Ts. 

If  in  eq.  (5)  we  assume  a  constant  spiral,  and  give  to 
K  two  values  in  succession  and  subtract  one  resulting 
value  of  Ts  from  the  other,  we  shall  find  for  their  dif- 
ference, 

diff.  Ts  =  "h  (**"').  diff.  jf.          .  (8.) 
cos  fa 

Hence 

diff.  jf  =    .  COS*A       diff.  Tv          .  (9.) 
sin  (iA  —  j) 

Example.  When  J?'  =  rad.  6°  54'  curve,  n  =  8,  c  — 
22,  7^  =  408.646  ;  what  radius  will  make  Ts  =  406  with 
the  same  spiral  ? 

Eq.  (9)     diff.  T,=  2.646  log  0.422590 

•3- A,   21°  log  COS  9.970152 

(-3- A  —  s),  15°  a.  c.  log  sin  0.587004 

.'.  diff.  1?  9-544  0.979746 

*'  6°  54'  830.876 

.  *.  Required  radius  —  821.332,  or  6°  58'  49"  curve. 

Remark.  Care  must  be  taken  to  observe  whether  in 
thus  changing  the  value  of  7?',  the  value  of  Z>',  the  de- 
gree of  curve,  is  so  far  changed  as  to  require  a  different 
spiral  according  to  the  rule  for  the  selection  of  spiral^ 
§  10.  Should  this  be  the  case  (which  is  not  very  likely), 
we  may  adopt  the  new  spiral,  and  proceed  with  a  new 
calculation  as  before. 

1 9-  Given  :  a  circular  curve  with  spirals  joining  two 
tangents,  to  find  the  external  distance  £.  —  VH, 
Fig-  5- 


20  THE    RAILROAD    SPIRAL. 

Let  SL  be  the  spiral,  LH  one-half  the  circular  curve, 
and  O  its  centre. 

Then  VH  =  VG  +  GO  -  OH. 

But      VG  =  ^T^r  —  -  ~~TT  >  and  in  the  triangle 

cos  VGN       cos  £A 

rni    rn       T  n  sin  OLI   --  &    cos  s 
GOL,  GO  =  LO  —. — ^-^7^  —  Jt 

sin  LGO  ( 


cos 
x  _;    cos  s 


s      --  1—  -        T>     •     •        . 

COS-^A  COS^A 

or  for  computation  without  logarithms 

^  f  ^  (cos  ^  —  cosJ-A  )  /       x 

^  "cosfA" 

Example.  Let  Z>f  =  7°  20',  A  =  42°,  and  for  the 
spiral  let  n  —  9,  ^  =  23,  giving  ^  =  7°  30',  and  for 
(0  +  i),  .A  -  7°  T5'  °4"- 

Eq.  (10)    a;  log         0.978743 

^A    21°  a.  c.  log  cos  0.029848 

10.200  1.008591 

K     7°  20'  log         2.893118 

s       7°  30'  log  cos  9.996269 

-JA2I0  a.  c.  log  cos  0.029848 

830.300  2.919235 


sum    840.500 
./?'     7°  20'  781.840 

£s  58.660 


ELEMENTARY    PROBLEMS.  21 

20.  Given  :  The  angle  A  at  the  vertex  and  the  dis- 
tance VH  =  £„  to  determine  the  radius  R'  of  a 
circular  curve  with  spirals  connecting  the  tangents 
and  passing  through  the  point  H.  Fig.  5. 

Solving  eq.  (n)  for  R'  we  have 

K  _  E,  cos  -j-  A  —  x 

cos  s  —  cos  4-  A  •  \     •) 

But  as  this  expression  involves  x  and  s  of  a  spiral  de- 
pendent on  the  value  of  R'  we  must  first  find  R'  approxi- 
mately, then  select  the  spiral,  and  finally  determine  the 
exact  value  of  R'  by  eq.  (12).  The  radius  R  of  a  simple 
curve  passing  through  the  point  H  is  a  good  approxima- 
tion to  R '.  It  is  found  by  eq.  (27)  Field  Engineering: 


R 


exsec  -J-A  ' 


or  the  degree  of  curve  D  may  be  found  by  dividing  the 
external  distance  of  a  i°  curve  for  the  angle  A  by  the 
given  value  of  Es.  But  evidently  the  value  of  D'  will 
be  greater  than  Dy  and  we  may  assume  D'  to  be  from 
10'  to  ic  greater  according  to  the  given  value  of  A,  the 
difference  being  more  as  A  is  less.  We  now  select  from 
Table  III.  a  value  of  DK  suited  to  D'  so  assumed,  and 
corresponding  at  the  same  time  to  any  desired  length  of 
spiral.  Since  Ds  so  selected  corresponds  to  (n  4-  i)  we 
take  the  values  of  n  and  x  from  the  next  line  above 
Ds  in  the  table,  find  the  value  of  s  from  Table  IV.,  and 
by  substituting  them  in  eq.  (12)  derive  the  true  value  of 
R1  for  the  spiral  selected. 

Example.  Let  A  —  42°  and  Es  =  70,  to  find  the  value 
of  R'  with  suitable  spirals. 

From  table  of  externals  for  i°  curve,  when  A  =  42° 
E  —  407--64,  which  divided  by  70  gives  5°.823  ;  or  D  = 


22  THE    RAILROAD    SPIRAL. 

5°  50'.  Assume  D'  say  20'  greater,  giving  D'  —  6°  10' 
approx.  If  we  desire  a  spiral  about  300  feet  long  we 
find,  Table  III.,  n  =  10,  c  =  30,  and  for  (n  -f  i)  Ds  — 
6°  06'  49".  For  72  =  10,  s  =  9°  10'. 

Eq.  (12)  cos^A,  21°  -93358 

^  7° 


65-35o6° 
x  16.768 

48.5826       log  1.686481 
cos  s  9°  10'     .98723 

COsiA     21°      .93358  .05365       log    8.729570 

.*.  R'  =  rad.   (say)  6°  20'  curve.  905.55   2.956911 

Proof.  Take  the  exact  radius  of  a  6°  20'  curve  and 
the  above  spiral  and  calculate  Ea  by  eq.  (10)  or  (n). 
We  shall  obtain  Es  =  69.97.  Again  :  if  we  desire  a 
spiral  of  200  feet,  we  find,  Table  III.,  n  =  8,  c  =  25,  and 
for  (n  +  i)  D,  —  6°,  and  by  eq.  (12)  R'  —  rad.  of  (say) 
6°  02'  curve  ;  and  by  way  of  proof  we  find  E,  =  69.96. 

Again  :  if  we  desire  a  spiral  of  about  400  feet,  we  find, 
Table  III.,  n  =  12,  c  =  33,  s  —  13°,  and  for  (n  -f  i) 
D,  —  6°  34'  07".  Hence  by  eq.  (12)  R'  —  rad.  of  (say) 
6°  50'  curve.  By  way  of  proof  we  find  eq.  (TO)  E,  = 

Remark.  It  is  thus  evident  that  a  variety  of  curves 
with  suitable  spirals  will  satisfy  the  problem,  but  D'  is 
increased  as  the  spiral  is  lengthened — for  in  the  ex- 
ample, with  a  200  ft.  spiral,  D'  —  6°  02' ;  with  a  300  ft. 
spiral,  D'  =  6°  20';  and  with  a  396  ft.  spiral,  I)'  — 
6°  50'.  Therefore  the  length  of  spiral,  as  well  as  the 
value  of  A,  must  be  considered  in  first  assuming  the 
value  of  I}'  as  compared  with  D  of  a  simple  curve. 


ELEMENTARY    PROBLEMS.  23 

21.  In  case  the  value  of  R  ',  as  calculated  by  eq.  (12), 
should  give  a  value  to  D'  inconsistent  with  the  spiral 
assumed,  we  may  easily  ascertain  by  consulting  the 
table  what  spiral  will  be  suitable.  Choosing  a  spiral  of 
the  same  number  of  chords,  but  of  a  different  chord- 
length  c,  we  may  calculate  R'  (a  new  value)  as  before  ; 
or  the  work  may  be  somewhat  abbreviated  by  the  fol- 
lowing method  : 

Given  :  a  change  in  the  value  of  x^  eq.  (12)  to  find  the 
corresponding  change  in  the  value  of  R'\  n  being  con- 
stant. 

If  the  values  of  Es,  A,  and  s  remain  unchanged,  we 
find,  by  giving  to  x  any  two  values,  and  subtracting  one 
resulting  value  of  R'  from  the  other, 


-  - 

COS  S  —  COS  -§•  A 


that  is,  R'  increases  as  x  decreases,  and  the  differences 

bear  the  ratio  of  -  —  =  —  . 

cos  s  —  cos  YA 

Example.   Let  A  =  42°,  Es  =  70,  and  for  the  spiral 

let  n—  10,   c  ~  30,  s  =  9°  10',  as  in  the  last  example, 

giving  R'  =  905.55  ;  to  find   the  change  in  R'  due  to 
changing  c  from  30  to  29. 

Eq.  (13)  for  c  —  30,  x  =  16.768 
for  c  —  29,  x  =  16.209 

diff.  x  .559  log  9.7474 

cos  s  —  COS^A  (as  before)    .05365  log  8.7296 

.'.  diff.  R'  10.42  1.0178 

old  value  9°5-55 


.  *.  new  R'  9T5-97  -&'  —  (say)  6°  J6', 


24  THE    RAILROAD    SPIRAL. 

which  agrees  well  with  Ds  —  6°  19'  29"  for  (n  4-  i)  in 
the  new  spiral. 

If  we  prove  this  result  by  calculating  the  value  of 
Es  for  these  new  values  by  eq.  (10)  we  shall  find  E,  = 
69.93. 

The  slight  discrepancy  between  these  calculated 
values  of  Es  and  the  original  is  due  solely  to  assuming 
the  value  of  D'  at  an  exact  minute  instead  of  at  a 
fraction. 


SPECIAL    PROBLEMS. 


CHAPTER  IV. 


SPECIAL    PROBLEMS. 

22.  Given  :  two  tangents  joined  by  a  simple  curve,  to 
find  a  circular  arc  with  spirals  joining  the  same  tan- 
gents, that  will  replace  the  simple  curve  on  the 
same  ground  as  nearly  as  may  be,  and  preserve  the 
same  length  of  line.  Fig.  6. 

To  fulfill  these  conditions  it  is  evident  that  the  new 
curve  must  be  outside  of  the  old  one  at  the  middle 
point  H,  since  the 
spirals  are  inside 
of  the  simple 
curve  at  its  tan- 
gent points  ;  also, 
the  radius  of  the 
new  "curve  must 
be  less  than  that 
of  the  old  one, 
otherwise  the  cir- 
cle passing  out-  ox 
side  of  H  would 
cut  the  given  tan- 
gents. 

Let  SV,  Fig.  6 
be    one    tangent, 

and  V  the  vertex.  FlG-  6- 

Let  AH  be  one  half  the  simple  curve,  and  O  its  centre. 
Let  SL  be  one  spiral,  LH'  one  half  the  new  circular 


26  THE    RAILROAD    SPIRAL. 

arc,  and  O'  its  centre.  Draw  the  bisecting  line  VO,  the 
radii  AO  =  R  and  LO'  =  R ',  and  the  perpendicular 
LM  =  x.  Then  MS  =y.  Produce  the  arc  H'L  to  A' 
to  meet  the  radius  O'A'  drawn  parallel  to  OA,  and  let 
| A  =  the  angle  AOH  =  A'O'H'.  Let  s  =  the  angle 
A 'OX  =  the  angle  of  the  spiral  SL.  Let  h  —  the  radial 
offset  HH'  at  the  middle  point  of  the  curve.  Draw 
O'N  and  LF  perpendicular  to  OA,  LF  intersecting  O'A 
at  I. 

a.  To  find  the  radius  R1  of  the  new  arc  LH'  in  terms 
of  a  selected  spiral  SL. 

We  have  from  the  figure  AO  =  ML  -f  FN  +  NO. 
But  AO  =  R,  ML  =  x,  FN  =  LO'  cos  s  =  R'  cos  s 
and  NO  =  O'O  cos  £  A  =  (OH'  -  O'H')  cos  £  A  = 
(h  +  R  —  R'}  cos  \  A  ;  and  substituting  we  have 

R  =  x  +  R'  cos  s  +  (h  +  R  —R')  cos  £  A  .      (14.) 
whence 

,  _        R  vers  ^  A  h  +  cos  \  A  4-  x      .      , 

cos  s  —  cos  £  A  cos  ^  —  cos  \  A 

It  is  found  in  practice  that  h  bears  a  nearly  constant 
ratio  to  x  for  all  cases  under  the  conditions  assumed  in 

this  problem.  Let  k  =  the  ratio  —  —  and  the  last  equa- 
tion may  be  written 

,  _       R  vers  \  A        __  (/EcosjA   +  i)x        ,  ,  x 
cos  s—  cos  4  A          cos  s  —  cos  -J  A 

which  gives  the  radius  of  the  new  arc  LH'  in  terms  of 
s,  ~v  and  k. 


SPECIAL    PROBLEMS.  27 

b.    To  find  the  offset  h  =  HH'  : 
From  eq.  (14)  we  derive 

h  cos  \  A  —  R  (  i  —  cos  4-  A  )  —  R  (  i  —  vers  s)  4- 

R'  cos  Y  A  —  x 
=  jB(i  -  cos|  A)  —  ^'O-  cosiA)-h 

-#'  vers  s  —  x 

—  (R  —  R')  vers  J  A   +  -#'  vers  .$•  —  #. 
Hence 


*g=X«  -  *')  exsec  4-  A  +  --  —       (17.) 

cos  i  A        cos  i  A 

which  gives  the  value  of  h  in  terms  of  s,  x  and  -#'. 


C.    70  /«//  ///^  ^//^  of  d  —  AS  : 

We  have  from  the  figure  SM  =  SA  +  NO'  +  IL. 
But  SM=j;,  SA  =  4  NO'=  OO;  sin  |  A  and  IL  = 
LO'  sin  s,  and  by  substitution, 

y  —  d  +  (h  +  R  —  R')  sin  -J-  A  +7?'  sin  j. 
Hence 

^=7  ~  [(;/  +  R  —R')  sin|  A  +  ^'sin^]   (18.) 
\ 

which  gives  the  distance  on  the  tangent  from  the  point 
of  curve  A  to  the  point  of  spiral  S. 

d.   To  compare  the  lengths  of  the  new  and  old  lines  : 
SAH  =  SA  +  AH  =  d+  100^-,     .     .(19.) 


in  which  D  is  the  degree  of  curve  of  AH  ; 

SLH'  =  SL  +  LH'  =  n  .  c  +  100  *£ 
in  which  D'  is  the  degree  of  curve  of  LH'. 


28  THE    RAILROAD    SPIRAL. 

If  the  spiral  and  arc  have  been  properly  selected,  the 
two  lines  will  be  of  equal  length  or  practically  so. 

The  last  two  equations  assume  the  circular  curves  to 
be  measured  by  100  foot  chords  in  the  usual  manner, 
but  when  the  curves  are  sharp  it  is  often  desirable  that 
they  should  agree  in  the  length  of  actual  arcs,  especially 
where  the  rail  is  already  laid  on  the  simple  curve.  For 
this  purpose  we  use  the  formulae 

SAH(arc)  =  </+*.  A. ~         •  •  (21.) 


SLH'  (arc)  =  n  .c  +  R          -  j    ~  (22.) 


in  which  the  angle  is  expressed  in  degrees  and  decimals. 
If  the  odd  minutes  in  the  angle  cannot  be  expressed  by 
an  exact  decimal  of  a  degree,  the  angle  should  be  re- 
duced to  minutes,  and  the  divisor  of  ft  changed  from 
180  to  10800. 


The  value  of—   -  is  .0174533  log  8.241877 

IoO 


it 

—- —  is  .00029089  :    6.463726. 

10800 

The  length  of  spiral  is  given  by  chord  measure  in  the 
last  equations,  since  the  chords  are  so  short  and  subtend 
such  small  angles  that  the  difference  between  chord  and 
arc  is  not  material  to  the  problem. 

e.  To  select  a  spiral  in  a  given  case,  we  require  to 
know  approximately  the  value  of  D',  and  to  select  the 
spiral  (n  .  c)  such  that  the  value  of  Ds  for  (n  +  i)  shall 
not  differ  greatly  from  the  value  of  D '.  To  aid  in  find- 


SPECIAL   PROBLEMS.  29 

ing  approximate  values  of  D'  and  k,  Table  V.  has  been 
prepared  for  curves  ranging  from  2°  to  16°  and  central 
angles  (A)  ranging  from  10°  to  80°. 

Assume  s  at  pleasure  (less  than  i  A ),  which  fixes  the 
value  of  n.  Then  inspect  Table  V.  opposite  n  for 
values  of  D  and  A  next  above  and  below  the  values  of 
D  and  A  in  the  given  problem,  and  by  inference  or  in- 
terpolation decide  on  the  probable  values  of  k  and  D'. 
Then  in  Table  III.  select  that  value  of  c  which  gives 
Ds  for  (n  -h  i)  most  nearly  agreeing  with  D' .  Now 
calculate  R'  by  eq.  (16),  and  as  this  will  usually  give 
the  degree  of  curve  D'  fractional,  take  the  value  of 
D'  to  the  nearest  minute  only,  and  assume  the  corre- 
sponding value  of  R'  as  the  real  value  of  R'.  A  table 
of  radii  makes  this  operation  very  simple. 

But  should  it  happen  that  D'  differs  too  widely  from 
from  Ds(n  +  ^  to  make  an  easy  curve,  increase  or  di- 
minish the  chord-length  c  by  i,  thus  giving  a  new  value 
to  x  in  eq.  (16),  and  also  a  new  value  of  J}s(n  +  I) 
with  which  to  compare  the  resulting  D'.  In  changing 
x  only  the  last  term  of  eq.  (16)  is  affected,  and  the  first 
term  does  not  require  recalculation. 

f.  When  the  value  of  R1  is  decided,  substitute  it  in 
eq.  (17)  and  calculate  h.  But  if  it  happens  that  the 
value  of  R'  selected  differs  not  materially  from  the  result 
of  eq.  (16),  we  have  at  once  h  =  kx  ;  or  in  case  the  value 
of  R'  is  changed  considerably  from  the  result  of  eq.  (16), 
the  corresponding  change  in  h  will  be 

.      .......    .  cos  s  —  cos  -J-A   ,.„    D,      ,     1X 

diff.  h  —  -  —  diff.  R ,  .  (22t) 

cosf  A 

which  may  therefore  be  applied  as  a  correction  to  h  —  kx, 
and  we  thus  avoid  the  use  of  eq.  (17).  Eq.  (22^)  is  de- 


30  THE    RAILROAD    SPIRAL. 

rived  from  eq.  (15)  by  supposing  h  to  have  any  two 
values,  and  subtracting  the  resulting  values  of  R'  from 
each  other.  Note  that  h  diminishes  as  R!  increases,  and 
vice  versa. 

When  R'  and  h  are  found,  proceed  to  find  d  by  eq. 
(18),  and  the  length  of  lines  by  eq.  (19),  (20),  or  by 
(21),  (22),  as  may  be  preferred.  But  to  produce  equal- 
ity of  actual  arcs,  k  must  be  a  little  greater  than  when 
equality  by  chord-measure  is  desired. 

Should  the  lines  not  agree  in  length  so  nearly  as  de- 
sired, a  change  of  one  minute  ±  in  the  value  of  D1 
may  produce  the  desired  result,  but  any  such  change 
necessitates,  of  course,  a  recalculation  of  h  and  d. 

The  values  of  k  in  Table  V.  appear  to  vary  irregu- 
larly. This  is  due  to  the  selection  of  D'  to  the  nearest 
minute,  and  also  to  the  choice  of  spiral  chord-lengths, 
c,  not  in  an  exact  series.  The  reader  is  recommended 
to  supplement  this  table  by  a  record  of  the  problems  he 
solves,  so  that  the  values  of  R'  and  k  may  be  approxi- 
mated with  greater  certainty. 

Example.  Given  a  6°  curve,  with  a  central  angle  of 
A  —  50°  12',  to  replace  it  by  a  circular  arc  with  spirals, 
preserving  the  same  length  of  line.  Assume  s  =  7°  30' 
giving  n  =  9. 

Since  6°  is  an  average  of  4°  and  8°,  while  50°  12'  is 
nearly  an  average  of  40°  and  60°,  we  examine  Table  V. 
under  4°  curve  and  8°  curve,  and  opposite  A  —  40° 
and  60°  on  the  same  line  as  s  =  7°  30',  and  take  an 
average  of  the  four  values  of  Z>s(n  +  r),  thus  found; 
also  of  the  four  values  of  k  ;  we  thus  find  approx.  k  = 
.0885,  and  D'  =  6°  18'  ±.  Now  looking  in  Table  III^ 
opposite  n  —  9,  we  find  that  when  c  =  26,  Ds  (n  +  i)  — 
6°  24'  48",  we  therefore  assume  c  =  26,  and  proceed  to 
calculate  R'  by  eq.  (16). 


SPECIAL    PROBLEMS.  31 

Eq.  (16)  cos  j  7°  30'  -99X44 

COS^A          25°  06'  -9055^ 

.08587  a.  c.  log  1.066159 

R                  6°                                               log  2.980170 

vers^-A      25°  o6r                                        log  8.975116 

1050.6           log  3.021445 

cos  s  —  cos  -JA                •                  a.  c.  log  1.066159 

I  '-f  y^COS  4- A  =  1. 080  0.033424 

x  1.031989 

135-4  2.131572 

.'.  R'  (say  6°  16')  9Z5-2 

Eq.(i7)     JK6°  955-366 

R'  6°  1 6'      914-75° 

(R  —  R')                    40.616                     log  1.608697 

exsec  ^A  25°  06'                                        log  9.018194 

4.235           log  0.626891 

R'                6°  16'                                       log  2.961303 

vers  s  ,       7°  30'                                        log  7.932227 

cos  -}A      25°  06'                               a.  c.  log  0.043079 

8.642           log  0.936609 

12.877 

log  1.031989 

i      25°  .06'                               a.  c.  log  0.043079 

11.887  1-075068 

.*.   h  0.990 

Eq.  (18)  (R  -  R')  40.616 

41.606           log  1.619156 

sin  ^-A       25°  06'                                        log  9.627570 

17.649            log  1.246726 


32 


THE    RAILROAD    SPIRAL. 


Rr      6°  16' 

sin  s  7°  30' 

119.399 

log 
log 

log 

2.961303 
9.115698 

2.077001 

137.048 

y 

.'.  d 

n      (  r  r\  I      -  .    *  — 

233-579 

A  T  R     1  TJ 

.'.   SAH 


514.864 


Eq.  (20)  (^A  —  s)  —  1056'  X  100 
D'  376' 


n  . c  9  x  26 

.--.  SLH' 
Difference 

h 

actual  k  =  —  =  0.092 
x 


280.851 
234- 

SM-^i 
-.013 


log     5.023664 
log     2.575188 


log     2.448476 


Comparison  of  actual  arcs. 


Eq.  (21)  25.1°  log  1.399674 

i°  log  8.241877 

R      6°  log  2.980170 


418.525    log  2.621721 
96-531 


Eq.  (22)  17.6°  log  1.245513 
i°  log  8.241877 
R'  6°  i67  log  2.961303 

280.991  log  2.448693 
n.c     234. 


5T4-991 
Difference  =  —  0.065 


SPECIAL    PROBLEMS. 


33 


23»  Given  :  a  simple  curve  joining  two  tangents,  to 
move  the  curve  inward  along  the  bisecting  line  VO 
so  that  it  may  join  a  given  spiral  without  change 
of  radius.  Fig.  7. 

Let  SL  be  the  given 
spiral,  AH  one-half  of  the 
given  curve,  and  HL  a 
portion  of  the  same  curve 
in  its  new  position,  and 
compounded  with  the 
spiral  at  L. 

To  find  the  distance 
h  =  HH'  =  OO7  : 

Since  the  new  radius  is 
equal  to  the  old  one,  or 
^?'=  R,  we  have  from  eq. 
(17)  by  changing  the  sign 
of  h,  since  it  is  taken  in  the  opposite  direction, 

x  —  R  vers  s 


COS 


To  find  the  distance  d  =  AS  : 

Changing  the  sign  of  h  in  eq.  (18)  and  making  R!  — 
R  we  have 

d  —  y  —  (R  sin  s  —  h  sin  -J  A)      •     •     •     •   (24-) 

This  problem  is  best  adapted  to  curves  of  large 
radius  and  small  central  angle. 

Example.  Given,  a  curve  D  =  i  °  40'  and  A  — 
26°  40',  and  a  spiral  s  =  i°,  n  =  3,  and  c  =  40,  to  find 
//  and  d  and  the  length  LH'. 

Eq.  (23)  R  i°  40'  log  3-5363 

vers  s        i°  log  6. 1 82 7 

cos  i  A  13°  20'  a.  c.  log  0.0119 


34  -  THE    RAILROAD    SPIRAL. 

•538         log     9-7309 


x  log     9.9109 

cos-J-A*  a.  c.  log     0.0119 


.837  9.9228 


.'.//  .299 

Eq.  (24)       R  i°  40'  log     3.536289 

sin     s  iu  "       8.241855 

59.999  x-778l44 


-299  log     9-4757 

sin  4  A  13    20  «       9.3629 

.069  8.8386 


59-93° 
y  119.996 


.  * .  d  60.066 

H'O'L  ^  (|  A  -  s)  =  12°  20'      .-.      H'L  =  740  feet. 

24.  Given,  a  simple  curve  joining  two  tangents,  to 
compound  the  curve  near  each  end  with  an  arc 
and  spiral  joining  the  tangent  without  disturbing  the 
middle  portion  of  the  curve.  Fig.  8. 

Let  H  be  the  middle  point  of  the  given  curve,  Q  the 
point  of  compounding  with  the  new  arc,  and  L  the 
point  where  the  new  arc  joins  the  spiral  SL. 

Let  s  =  the  spiral  angle,  and  let  0  =  AOQ.  Now  in 
this  figure  AOQS  will  be  analogous  to  AOH'S  of  Fig. 6, 
if  in  the  latter  we  suppose  H'  to  coincide  with  H  or 
//  =  o.  If,  therefore,  in  eq.  (15)  we  write  0  for  -J-  A  and 
make  //  =  o,  we  have  for  the  new  radius  O'Q, 

,  _  R  vers  °  —  x .  . 

~  cos  s  —  cos  0'  ' 


SPECIAL    PROBLEMS. 


35 


in  terms  of  0  and  the 
spiral  assumed.  But 
as  the  value  of  D' 
resulting  is  likely  to 
be  fractional  and 
must  be  adhered  to, 
it  is  preferable  to  as- 
sume jR'  a  little  less 
than  R,  select  a  suit- 
able spiral  and  cal- 
culate the  angle  0. 
Resolving  eq.  (17) 
after  making  h  =  o 
and  replacing  \  A 
by  0,  we  have 


FIG.  8. 


vers  0  := 


•r-JT' 


vers  s 


(26.) 


The  angle  0  so  found  must  be  less  than  -3-  A ,  and  in- 
deed for  good 'practice  should  not  exceed  3-  A.  If  too 
large,  0  may  be  reduced  by  assuming  a  smaller  value  of 
^',  and  repeating  the  calculation  with  a  suitable  spiral. 
Otherwise  it  will  be  preferable  to  use  one  of  the  forego- 
ing problems  in  place  of  this.  This  problem  is  specially 
useful  when  the  central  angle  is  very  large. 

To  find  the  distance  d  —  AS,  we  have  only  to  write 
0  for  4  A  and  make  h  =  o  in  eq.  (18),  whence 

d—y  -  \(R  -  J?')  sin  0  +  R'  sin^]    .    .    .    (27.) 

Example.  Given  a  curve  D  ~  2°  30',  A  =35°,  to 
compound  it  with  a  curve  D'  =  2°  40'  and  a  spiral  s  = 
2°  30',  n  =  5,  <•  =  37. 


THE    RAILROAD    SPIRAL. 


Eq.  (26)  R   2°  30'  2292.01 
R'  2°  40'  2148.79 


R-  R1 

X 


143.22 


R-  R' 

vers  s    2°  30' 

o          r 
2      40 


R' 


.-.  versO  6°  28'  30" 

Eq.  (21}R-R' 

sin  <*       6°  28'  30" 


R' 

sin  s 


2U  40' 
2°  30' 


.-.  d 

AH 


log  2.156004 
log  0.471203 


0020663  log  8.315199 

a.  c.  log  7.843996 

log  6.978536 

*  log  3-332I93 

log  8. 154725 


.014280 
.006383 


16.151 


93.729 

109.880 
184.962 


log  2.156004 
9.052192 


1.208196 

3-332I93 
8.639680 

1.971873 


775.082 

SL,    —n.c—  185.00 

LQ,    0  -  s  =    3°  58'  30"  149.06 
QH,  i  A  —  0  =  n°  oi'  30"  441.00  775-°6° 


Difference 


—  .022 


SPECIAL    PROBLEMS. 


37 


25.  Given:  a  compound  curve  joining  two  tan- 
gents, to  replace  it  by  another  with  spirals,  pre- 
serving the  same  length  of  line.  Fig.  9. 

Let  A  2  =  AO2P, 
the  angle  of  the  arc 
AP,  and  A,  = 
PdB,  the  angle  of 
the  arc  PB.  Let 
^2  —  A  O2,  and 
R,  =  BO,. 

Adopting  the 
method  of  §  22, 
the  offset  h  must 
be  made  at  the 
point  of  compound 
curve  P  instead  of 
at  the  middle  point. 
Cons  idering  first 
the  arc  of  the 
larger  radius  AO2, 
the  formulae  of  §22 
will  be  made  to 


FIG.  9. 


apply  to  this  case  by  writing    A  2  in  place  of  \  A  ,  and 
Ri  in  place  of  R,  whence  eq.  (16) 


vers  A  g 


cos  s  —  cos  A  2 


(k  cos  A2  +  i)  x 
cos  s  —  cos  A  2 


, 


and  eq.  (17) 

7  /  r>  r>  f\  A         ,     •£*    vers  s  X  t         \ 

h  —  (R^  —  R^)  exsec  A2  +  -  (20.) 

COS    A  2  COS    A  2 

and  eq.  (18) 

d=y.  -  [(k  +  R,  -  RJ)  sin  A  2  +  R*  sin  s]  .  .    (30.) 


38  THE    RAILROAD    SPIRAL. 

But  in  considering  the  second  arc  PB,  we  must  retain 
the  value  of  //  already  found  in  eq.  (29)  in  order  that 
the  arcs  may  meet  in  P'.  We  therefore  use  eq.  (15) 
which,  after  the  necessary  changes  in  notation,  becomes 

n  ,          Rl  vers  A  l  h  cos  A  ,  +  x  ,      x 


.  .          . 

cos  s  —  cos  A  j        cos  s  —  cos  A  ! 

which  value  of  lt\  must  be  adhered  to. 

The  spiral  selected  for  use  in  the  last  equation  is  in- 
dependent of  the  spiral  just  used  in  connection  with  J?./. 
It  should  be  so  selected  that  while  suitable  for  ^?/  its 

value  of  x  may  be  equal  to  —  as   nearly  as  may  be,   the 

K 

value   of  k  being  inferred   from  Table  V.  for  D'  and 

2    A!. 

Assuming  the  value  of  7?/  found  by  eq.  (31)?  even 
though  DI  be  fractional,  we  may  verify  the  value  of  h  by 


h  =  (JP,  -  ft)  exsec  A  ,  +      L  -  -  (32.) 

cos  A  j         cos  A  j 

and  then  proceed  to  find  d'  =  BS'  by 

dj  -y-  [(h  +  ^  -J?,')  sin  A,  •f.tf/sin.f]  (33.) 

Example.  Given  the  compound  curve  D±  ==  8°.,  A  ,  — 
29°  and  Z>2  =  6°,  A2  =  25°o6'  :  to  replace  it  by  an- 
other compound  curve  connected  with  the  tangents  by 
spirals. 

Considering  first  the  6°  branch  of  the  curve,  we  may 
assume  the  spiral  s  =  7°3o',  n  =  9,  c  =  26.  This  part 
of  the  problem  is  then  identical  with  the  example  given 
in  §  22,  by  which  we  find  h  —  .990  and  d  —  96.531. 

To  select  a  spiral  for  the  8°  branch,  having  reference 
at  the  same  time  to  this  value  of  h  ;  we  find  in  Table  V. 


SPECIAL    PROBLEMS.  39 

under  D  =  8°  and  opposite  A  —  2  A  l  =  58°  or  say  60°, 
that  the  given  value  of  h  falls  between  the  tabular 
values  of  7*  for  nc  =  9  x  20,  and  //^  =  io"x  22.  We  there- 
fore infer  that  the  spiral  ^  —  9x21  is  most  suitable  to 
this  case.  Adopting  this,  we  have 

Eq.  (31)  COS.T  7°3o'. 99144 

COS   A!  29°. 87462 


.11682  log  9. 0675 1 7  a>c-  l 
R,  8°  "    2.855385 

vers  A^0  "  9.098229 


769.302  "  2.886097 

h  cos     29°     .866 
x  8.694 

9.560  :<  0.980458 

cos  s  —  cos  A!  a.c.     "  0.932483 


8i.835 

"   1.912941 

\ 

"  1.481471 
"9-685571 

.-.  US  8°2o'3or                 687.467 
[.  (33)  (h  +  .#,)                          717.769 

30.302 
sin  A  !  29° 

14.691 

J?/  687.467 
sin  s     7°3of 

89-732 

'  1.167042 

"  2.837251 

9.  115698 

1.952949 

104.423 
188.660 

y 

•'.  d  84.237 


40  THE    RAILROAD    SPIRAL. 

For  the  methods  of  computing  the  lengths  of  lines, 
see  §  22. 

26.  Given  :  a  compound  curve  joining  two  tangents, 
to  move  the  curve  inward  along  the  line  PO2  so  that 
spirals  may  be  introduced  without  changing  the  ra- 
dii. Fig.  10. 

The  distance  h  =  PP'  is  found  for  the  arc  of  larger 


Fig.  10. 

radius  AO2  by  the  following  formula  derived  by  analogy 
from  eq.  (23): 

,  ___  x  —  R*  vers  s  .  ,      ^ 

cos  A2 

and  for  the  distance  d  =•  AS  we  have  analogous   to   eq. 
d  —  y—  (£z  sin,  —  /fcsin  A2)       .       (35.) 


SPECIAL    PROBLEMS.  41 

.  Now  the  same  value  of  /*,  found  by  eq.  (34)  must  be 
used  for  the  arc  PB,  and  a  spiral  must  be  selected  which 
will  produce  this  value.  To  find  the  proper  spiral,  we 
have  from  eq.  (34)  after  changing  the  subscripts, 

x  —  Rl  vers  s  +  h  cos  A  ^     .     .     (36.) 

The  last  term  is  constant.  The  values  of  x  and  s  must 
be  consistent  with  each  other,  and  approximately  so  with 
the  value  of  R^.  Assume  s  at  any  probable  value,  and 
calculate  x  by  eq.  (36).  Then  in  Table  III.  look  for 
this  value  of  x  opposite  n  corresponding  to  j,  and  note 
the  corresponding  value  of  the  chord-length  c.  Com- 
pare Ds  of  the  table  with  Z>i  and  if  the  disagreement  is 
too  g^eat  select  another  value  of  s  and  proceed  as  be- 
fore. 

The  term  JR^  vers  s  may  be  readily  found,  and  with 
sufficient  accuracy  for  this  purpose,  by  dividing  the  value 
of  R  i°  versj  Table  IV.  by  Z>lm  If  the  calculated  value 
of  x  is  not  in  the  Table  III.,  it  may  be  found  by  inter- 
polating values  of  c  to  the  one  tenth  of  a  foot,  since  for 
a  given  value  of  s  or/n  the  values  of  x  and  y  are  pro- 
portional to  the  values  of  c. 

When  the  proper  spiral  has  been  found  and  the  value 
of  c  determined,  it  only  remains  to  find  the  value  of  d  = 
BS'by 

d  —  y  —  (Ri  sin  s  —  h  sin  A  i),     .     (37.) 

in  which  the  value  of  y  will  be  taken  according  to  the 
values  of  c  and  s  just  established. 

Example.  Given:  Z>2  —  i°4o',  A2  =  13° 20',  Z>1  =  3°, 
and  A  =  22°4o',  to  apply  spirals  without  change  of 
radii.  Fig.  10. 

Assume  for  the  i°  40'  arc  the  spiral  s  —  i°,  n  =  3, 
c  —  40.  This  part  of  the  problem  is  then  identical  with 
the  example  given  in  §  23,  from  which  we  find  h  =  0.299. 


THE    RAILROAD    SPIRAL. 


For  the  second  part,  if  we  assume  s  —  i°  40',  n  =  4, 
and  find  by  Table  IV.  ^  vers  s  =  — ^  =    0.808,     we 

o 

have  by  eq.  (36) 

x  =  0.808  -f  0.277  —  1.085, 

the  nearest  value  to  which  in  Table  III.  is  under  c  = 
25,  giving Ds  =2°  40',  or  for  (n  +  i),  D»  =  3°  20',  which 
is  consistent  with  Dl  —  3°.  By  interpolation  we  find 
that  our  value  of  x  corresponds  exactly  to  c  =  24.85, 
n  =  4,  and  therefore  the  spiral  should  be  laid  out  on  the 
ground  by  using  this  precise  chord. 

In  order  to  find  d  =  BS'  we  first  find  the  value  of  y 
by  interpolation  for  c  =24.85,  when  by  eq.  (37)  we^  have 

*T=  99-391  -  (55-554  ~  0.115)  =  43-952- 
27.  Given  :  a  compound  curve  joining  two  tan- 
gents, to    introduce  spirals  without  disturbing 

the    point    of 
^^ 

B 


compou  nd 
curvature    P. 

Fig.  ii. 

a.  The  radius 
of  each  arc  may 
be  shortened,  giv- 
ing two  new  arcs 
compounded  at 
the  same  point 
P.  Having  se- 
lected a  suitable 
spiral,  we  have 
for  the  arc  AP 
s  by  analogy  from 
eq-  (15),  since 


Fig. 


SPECIAL    PROBLEMS. 

_  Rv  vers  A  2  —  x 
cos  ^  —  cos  A  2  ' 


43 


(38.) 


and,  similarly,  after  selecting   another  spiral  for  the  arc 
PB, 

_  ^,  vers  A ,  —  x 

cos  s  —  cos  Aj  "     "     V39-J 

From  eq.  (18)  we  have  for  the  distance  AS, 

d~y—  \_(R*  —  RI']  sin  A  2  +  RJ  sin  s],  .  (40.) 

and  for  the  distance  BS', 

/ 
d  ~ y  —  [(XL  —  Ri')  sin  A!  +  RI   sin  s]  .   (41.) 

The  values  of 
DI  and  Dj  re- 
sulting from  eq. 
(39)  and  (40) 
must  be  adhered 
to,  even  though 
involving  a  frac- 
tion of  a  minute. 

b.  Either  arc 
may  be  again  com- 
pounded at  some 
point  Q,  leaving 
the  portion  PQ 
undisturbed,  as 
explained  in  §  24. 
Fig.  12. 

Let  e  =  the  an- 


Fig.    12. 


gle  AO2Q,  and  we  have  from  eq.  (26),  after  selecting  a 
suitable  spiral  and  assuming  .#./, 


vers  0  — 


vers  s 


44  THE    RAILROAD    SPIRAL. 

For  the  distance  AS,  we  have  from  eq.  (27) 

d  =  y  -  [(.£„  -  A')  sin  0  +  RJ  sin  j]    .     (43.) 

Similar  formulae  will  determine  the  angle  0  =  BOjQ' 
and  the  distance  BS'  for  the  other  arc  PB  in  terms  of  a 
suitable  spiral  :  thus, 

x  —R\   vers  s 


d—  y  -  [(^  -  ^/)  sin  0  +  ^/  sin  s]     .     (45.) 


The  method  a  may  be  adopted  with  one  arc  and  the 
method  b  with  the  other  if  desired,  since  the  point  P  is 
not  disturbed  in  either  case.  The  former  is  better 
adapted  to  short  arcs,  the  latter  to  long  ones. 

These  methods  apply  also  to  compound  curves  of 
more  than  two  arcs,  only  the  extreme  arcs  being  altered 
in  such  cases. 


FIELD    WORK.  45 


CHAPTER   V. 

FIELD     WORK. 

28.  HAVING  prepared  the  necessary  data  by  any  of 
the  preceding  formulae,  the  engineer*  locates  the  point  S 
on  the  ground  by  measuring  along  the  tangent  from  V 
or  from  A.  He  then  places  the  transit  at  S,  makes  the 
verniers  read  zero,  and  fixes  the  cross-hair  upon  the  tan- 
gent. He  then  instructs  the  chainmen  as  to  the  proper 
chord  c  to  use  in  locating  the  spiral,  and  as  they  meas- 
ure this  length  in  successive  chords,  he  makes  in  succes- 
sion the  deflections  given  in  Table  II.  under  the 
heading  "Inst.  at  S,"  lining  in  a  pin  or  stake  at  the  end 
of  each  chord  in  the  same  manner  as  for  a  circle. 

When  the  point  Las  reached  by  (n)  chords,  the  tran- 
sit is  brought  forward  and  placed  at  L  ;  the  verniers  are 
made  to  read  the  first  deflection  given  in  Table  II. 
under  the  heading  "  Inst.  at  n  "  (whatever  number  n  may 
be),  and  a  backsight  is  taken  on  the  point  S.  If  the 
verniers  are  made  to  read  the  succeeding  deflections,  the 
cross-hair,  should  fall  successively  on  the  pins  already 
set,  this  being  merely  a  check  on  the  work  done,  until 
when  the  verniers  read  zero,  the  cross-hair  will  define  the 
tangent  to  the  curve  at  L.  From  this  tangent  the  cir- 
cular arc  which  succeeds  may  be  located  in  the  usual 
manner. 

In  case  it  became  necessary  to  bring  forward  the  tran- 
sit before  the  point  L  is  reached,  select  for  a  transit- 
point  the  extremity  of  any  chord,  as  point  4,  for 


46  THE    RAILROAD    SPIRAL. 

example,  and  setting  up  the  transit  at  this  point,  make 
the  verniers  read  the  first  deflection  under  "  Inst.  at  4," 
Table  II.,  and  take  a  backsight  on  the  point  S.  Then, 
when  the  reading  is  zero,  the  cross-hair  will  define  the 
tangent  to  the  curve  at  the  point  4,  and  by  making  the 
deflections  which  follow  in  the  table  opposite  5,  6,  &c., 
those  points  will  be  located  on  the  ground  until  the 
desired  point  L  is  reached  by  n  chords  from  the  begin- 
ning S. 

The  transit  is  then  placed  at  L,  and  the  verniers  set 
at  the  deflection  found  under  the  heading  "  Inst.  at  n  " 
(whatever  number  //  may  be),  and  opposite  (4)  the  point 
just  quitted.  A  backsight  is  then  taken  on  point  4, 
and  the  tangent  to  the  curve  at  L  found  by  bringing  the 
zeros  together,  when  the  circular  arc  may  be  proceeded 
with  as  usual. 

29.  To  locate  a  spiral  from  the  point  L  running  toward 
the  tangent  at  S :  we  have  first  to  consider  the  number  of 
chords  (n)  of  which  the  spiral  SL  is  composed.  Then,, 
placing  the  transit  at  L,  reading  zero  upon  the  tangent 
to  the  curve  at  L,  look  in  Table  II.  under  the  heading 
"  Inst.  at  #,"  and  make  the  deflection  given  just  above 
o°  oo'  to  define  the  first  point  on  the  spiral  from  L 
toward  S  ;  the  next  deflection,  reading  up  the  page,  will 
give  the  next  point,  and  so  on  till  the  point  S  is 
reached. 

The  transit  is  then  placed  at  S  ;  the  reading  is  taken 
from  under  the  heading  "Inst.  at  S,"  and  on  the  line  n 
for  a  backsight  on  L.  Then  the  reading  zero  will  give 
the  tangent  to  the  spiral  at  the  point  S,  which  should 
coincide  with  the  given  tangent. 

If  S  is  not  visible  from  L,  the  transit  may  be  set  up 
at  any  intermediate  chord-point,  as  point  5,  for  example. 
The  reading  for  backsight  on  L  is  now  found  under  the 


FIELD    WORK.  47 

heading  "  Inst.  at  5,"  and  on  the  line  n  corresponding  to 
L  ;  while  the  readings  for  points  between  5  and  S  are 
found  above  the  line  5  of  the  same  table.  The  transit 
being  placed  at  S,  the  reading  for  backsight  on  5,  the 
point  just  quitted,  is  found  under  "  Inst.  at  S  "  and 
opposite  5,  when  by  bringing  the  zeros  together  a  tan- 
gent to  the  spiral  at  S  will  be  defined. 

30.  Since    the    spiral   is   located    exclusively   by   its 
chord-points,  if  it  be  desired  to  establish  the  regular  100- 
foot  stations  as  they  occur  upon  the  spiral,  these  must  be 
treated  asflusses  to  the  chord-points,  and  a  deflection 
angle  will  be  interpolated   where  a  station  occurs.      To 
find  the  deflection  angle  for  a  station  succeeding  any  chord- 
point :  the  differences  given  in  Table  II.  are  the  deflec- 
tions over  one  chord-length,  or  from  one  point  to  the 
next.     For  any  intermediate  station  the  deflection  will 
be  assumed  proportional  to  the   sub-chord,  or  distance 
of  the  station*  from  the  point.     We  therefore  multiply 
the  tabular  difference  by  the  sub-chord,  and  divide  by  the 
given  chord-length,  far  the  deflection  from  that  point  to 
the  station.     This  applied  to  the  deflection  for  the  point 
will  give  the  total  deflection  for  the  station. 

This  method  of  interpolation  really  fixes  the  station 
on  a  circle  passing  through  the  two  adjacent  chord- 
points  and  the  place  of  the  transit,  but  the  consequent 
error  is  too  small  to  be  noticeable  in  setting  an  ordinary 
stake.  Transit  centres  will  be  set  only  at  chord-points, 
as  already  explained. 

31.  It   is  important  that  the  spiral   should  join  the 
main  tangent  perfectly,  in  order  that  the  full  theoretic 
advantage   of  the  spiral  may  be  realized.     In  view  of 
this   fact,  and  on    account    of  the    slight  inaccuracies 
inseparable  from  field  work  as  ordinarily  performed,  it 
is  usually  preferable  to  establish  carefully  the  two  points 


48  THE    RAILROAD    SPIRAL. 

of  spiral  S  and  S'  on  the  main  tangents,  and  beginning 
at  each  of  these  in  succession,  locate  the  spirals  to  the 
points  L  and  L'.  The  latter  points  are  then  connected 
by  means  of  the  proper  circular  arc  or  arcs.  Any  slight 
inaccuracy  will  thus  be  distributed  in  the  body  of  the 
curve,  and  the  spirals  will  be  in  perfect  condition. 

32.  A  spiral  may  be  located  without  deflection  angles, 
by  simply  laying  off  in   succession  the  abscissas  y  and 
ordinates  x  of  Table  III.  corresponding  to   the  given 
chord-length  c.     The  tangent  EL  at  any  point  L,  Fig.  4, 
is  then  found  by  laying  off  on  the  main  tangent  the  dis- 
tance   YE  =  x  cot  s,   and  joining  EL.     In   using   this 
method  the  chord- length  should  be  measured  along  the 
spiral  as  a  check. 

33.  In  making  the  final   location  of  a  railway  line 
through  a  smooth  country  the  spirals  may  be  introduced 
at  once  by  the  methods  explained  in  Chapter  III.     But 
if  the  ground  is  difficult  and  the  curves  require  close  ad- 
justment to  the  contour  of  the   surface,  it  will  be  more 
convenient  to  make  the  study  of  the  location  in  circular 
curves,  and  when  these  are  likely  to  require  no  further 
alterations,  the  spirals  may  be  introduced  at  leisure  by 
the   methods   explained   in  Chapter  IV.     The    spirals 
should  be  located  before  the  work  is  staked  out  for  con- 
struction, so   that  the  road-bed  and  masonry  structures 
may  conform  to  the  centre  line  of  the  track. 

34,  When  the  line  has  been  first  located  by  circular 
curves  and  tangents,  a  description  of  these  will  ordi- 
narily suffice  for  right  of  way  purposes  ;  but  if  greater 
precision  is  required  the  description  may  include  the 
spirals,  as  in  the  following  example  : 

"  Thence  by  a  tangent  N.  10°  i5'E.,  725  feet  to  station 
1132  +  12;  thence  curving  left  by  a  spiral  of  8  chords, 
288  feet  to  station  1 135;  thence  by  a  4°  12'  curve  (radius 


FIELD    WORK.  49 

1364.5  feet),  666.7  feet  to  the  station  1141  +66.7;  thence 
by  a  spiral  of  8  chords  288  feet  to  station  1144  +  54.7 
P.T.  Total  angle  40°  left.  Thence  by  a  tangent  N.  29° 
45'  W.,"  &c. 

35.  When  the  track  is  laid,  the  outer  rail  should  re- 
ceive a  relative  elevation   at   the  point  L  suitable  to  the 
circular  curve  at  the  assumed  maximum  velocity.     Usu- 
ally the  track  should  be  level   transversly  at  the  point  S, 
but  in  case  of  very  short  spirals,  which  sometimes  can- 
not be  avoided,  it  is   well  to  begin  the  elevation  of  the 
rail  just  one  chord-length  back  of  S  on  the  tangent. 

36.  Inasmuch  as  the  perfection  of  the   line   depends 
on  adjusting  the  inclination  of  the  track  proportionally 
to  the  curvature,  and  in  keeping  it  so,  it  is  extremely  im- 
portant that  the  points  S  and  L  of  each  spiral  should  be 
secured  by  permanent  monuments  in  the  centre  of  the 
track,  and  by  witness-posts  at  the  side  of  the  road.  The 
posts  should  be  painted  and  lettered  so  that  they  may 
serve  as   guides   to   the   trackmen  in   their    subsequent 
efforts  to  grade  and  "line  up  "  the  track.     The  post  op- 
posite the  point  S  may  receive  that  initial,  and  the  post 
at  L  may  be  so  marked   and   also   should   receive   the 
figures  indicating  the  degree  of  curve. 

37.  The  field  notes  may  be  kept  in  the  usual  manner 
for  curves,  introducing  the  proper  initials  at  the  several 
points  as  they  occur.     The   chord-points  of  the  spiral 
may  be  designated  as  plusses  from  the  last  regular  sta- 
tion if  preferred,  as  well  as  by  the  numbers  i,  2,  3,  &c., 
from  the  point   S.     Observe    that   the   chord  numbers 
always  begin  at  S,  even  though  the  spiral  be  run  in  the 
opposite  direction. 


TABLE 


ELEMENTS   OF   THE   SPIRAL 


Inclina- 

Point 

Degree 
of  curve 

Spiral 
angle 

tion  of 
chord 

Latitude  of  each 
chord. 

Sum  of  the  lati- 
tudes, 

to  axis 

i 

n. 

Ds. 

s. 

of  Y. 

TOO  x  cos  Incl. 

?- 

0 

o°  oo' 

o°  oo' 

o°  oo' 

I 

10' 

10' 

05' 

99.99989423 

99.99989423 

2 

20' 

30' 

20' 

99.99830769 

199.99820192 

3 

30' 

1° 

45' 

99.99143275 

299.98963467 

4 

40' 

1°  40' 

1°  20' 

99.97292412 

399.96255879 

5 

50' 

2°  30' 

2°  05' 

99-93390007 

499  89645886 

6 

1° 

3°  30' 

3° 

99.8629535 

599.7594123 

7 

1°   10' 

4°  4«' 

4°  05' 

99.7461539 

699.5055662 

8 

1°  20' 

6° 

5°  20' 

99.5670790 

799.0726452 

9 

I'  30' 

7°  30' 

6°  45' 

99.3068457 

898.3794909 

10 

I°40' 

g°  10 

8°  2o| 

98.944164 

997-3236549 

ii 

I'  SO' 

11° 

10°  05' 

98.455415 

1095.779070 

12 

2° 

13° 

12° 

97.814760 

1193.593830 

13 

2°  10' 

15°  10' 

14°  05' 

96.994284 

1290.588114 

14 

2°  20' 

I7I  3° 

1  6°  20' 

95.964184 

1386.552298 

15 

2°  30' 

20° 

i8°45' 

94.693014 

1481.245312 

16 

2°  40' 

22°  40' 

21°  20' 

93-147975 

1574-393287 

17 

2°  50' 

25°  30 

24°  05' 

91.295292 

1665.688579 

18 

3° 

28°  30'' 

27° 

89.100650 

1754.789229 

19 

3°  10' 

31°  40 

3o°  05' 

86.529730 

1841.318959 

20 

3°  20' 

35° 

33°  20' 

83.548730 

1924.867739 

Point. 

Log^  = 

Deflection  angle, 

«. 

log  tan  /. 

it 

I 

7.1626964 

o°  05'  oo.  'oo 

2 

7.5606380 

o°  12'  30.  'oo 

3 

7.831709! 

0°  23'  20.  '00 

4 

8.0377730 

o°  37'  29.  '99 

5 

8.2041217 

o°  54'  59-  '97 

6 

8.3436473 

i°  15'  49.  '90 

7 

8.4638309 

i°  39'  59-  '75 

8 

8.5694047 

2°  07'  29.  '45 

9 

8.6635555 

2°  38'   IS.  '90 

10 

8.7485340 

3°  12'  27.  '95 

OF   CHORD-LENGTH,  100. 


Departure  of 

Sum  of  the  depart- 

Logarithm, 

Logarithm, 

Point 

each  chord. 

ures, 

100  x  sin  Incl. 

X. 

logjj/. 

log  jr. 

n. 

0 

.1454441 

.1454441 

1.9999995 

9.1626960 

I 

.5817731 

.7272172 

2.3010261 

9.8616641 

2 

I.308Q593 

2.0361765 

2.4771063 

0.3088154 

3 

2.3268960 

4.3630725 

2.6020194 

0.6397924 

4 

3.6353009 

7.9983734 

2.6988800 

0.9030017 

5 

5-233596 

13.231969 

2.7779771 

1.1216244 

6 

7.120730 

20.352699 

2.8447911 

1.3086220 

7 

9.  29499  [ 

29.647690 

2.9025862 

.4719909 

8 

11-75374 

41.40143 

2.9534598 

.6170153 

9 

14.49319 

55.89462 

2.9988361 

.7473701 

10 

17.50803 

73.40265 

3.0397231 

.8657117 

n 

20.79117 

94.19382 

3.0768567 

.9740224 

12 

24.33329 

118.52711 

3.1107877 

2.0738177 

13 

28.12251 

146.64962 

3.1419362 

2.I6628II 

14 

32.14395 

.  178.79357 

3.1706269 

2.2523519 

15 

36.37932 

215.17289 

3.1971131 

2.3327875 

16 

40.  80649 

255.97938 

3.2215938 

2.4082049 

17 

45.39905 

301.37843 

3.2442250 

2.4791121 

18 

50.12591 

35L50434 

3.2651291 

2.5459307 

19 

54.95090 

406.45524 

3.2844009 

2.6090128 

20 

T    * 

Deflection  an- 

Point 

L°S  y  = 

gle, 

n. 

log  tan  i. 

t. 

II 

8.8259886 

3°49'56."39 

12 

8.8971657 

4°3o'43."95 

13 

8.9630300 

5°  14'  50."28 

14 

9.0243449 

6°  02'  I4."93 

15 

9.0817250 

6°52'57."3* 

16 

9.1356744 

7°46'56."7i 

17 

9.1866111 

8°  44'  I2."26 

IS 

9.2348871 

9°  44'  42.  "92 

19 

9.2808016 

10°  48'  27.  "44 

20 

9.3246119 

n°55'24."34 

TABLE    IT. 

DEFLECTION    ANGLES,  FOR    LOCATING    SPIRAL  CURVES    IN   THE 
FIELD. 

Rule  for  finding  a  Deflection. 

Read  under  the  heading  corresponding  to  the  point  at  which  the 
instrument  stands,  and  on  the  line  of  the  number  of  the  point 
observed. 


INSTRUMENT    AT    S. 

s  =  o. 

No.  of  Point, 

Deflection 

from 

Tangent, 

Difference 

of  Deflec- 

n. 

i. 

tion. 

0 

oo' 

I 

05 

05 

2 

3 
4 
5 
6 

7 

8 

9 

10 

ir 

12 
13 
14 
15 

16 

17 

18 

19 

20 

i° 

i 

2 

2 

3 
3 
4 
5 
6 
6 

7 
8 

9 

10 

ii 

12 

23 
37 
55 
15 
40 
07 
38 

12 

49 
30 
14 

02 
52 
46 

44 
44 

48 

55 

30" 

20 

30 
00 

50 

00 

29 
19 

28 

56 

.44 
SG 
15 

57 
57 

12 

43 
27 
24 

07 

10 

14 
17 

20 

24 
27 
30 

34 
37 
40 

44 
47 
50 

54 
57 
60 

63 
66 

30' 
50 
10 
30 
50 
10 

29 
50 
09 

28    - 

48 
06 
25 
42 

oo 

15 
31 

44 

57 

52 


TABLE  II. — DEFLECTION  ANGLES. 


INST.  AT  i.         s  =  o°  10'. 

INST.  AT  2.        s  =  o°  30'. 

No.  of    Deflection  'from 

Diff  .  of 

No.  of 

Deflection  from 

Diff.  of 

Point. 

aux.  tan. 

Deflection. 

Point. 

aux.  tan. 

Deflection. 

O 
I 

05' 
00 

05' 

0 

I 

17'  30" 
10 

7'  30" 

IO 

10 

2 

IO 

2 

00 

12    30' 

15 

3 
4 

5 
6 

7 
8 

9 

10 

ii 

12 
13 
14 
15 

16 

17 

18 

-  19 

20 

22    30" 

38    20 

57  3o' 

I9    20   00 

i     45   50 

2       15    00 

2     47  29 
3     23   18 
4    02  27 
4     44  55 
5     30  42 
6     19  47 
7     12  ii 
8     07  51 
9     06  49 
10    09  01 
ii     14  28 

12      23    08 

15    50 

19  10 

22    30 

25  50 
29  10 

32  29 

35  49 
39  09 
-    42  28 
45  47 
49  05 
52  24 
55  40 
58  58 

62    12 
65    27 

68  40 

3 
4 

5 
6 

8 
9 

10 

ii 

12 
13 
14 
15 

16 

17 

18 

19 

20 

15 

32  30 
53  20 
ic   17  30 
i     45  oo 
2     15  50 
2    49  59 
3     27  29 
4    08  18 
4     52  26 

5     39  54 
6     30  40 
7     24  44 
8     22  06 
9     22  45 
10     26  39 

ii     33  49 
12     44  12 

17  30 

20    50 

24  10 

27    30 
30    50 

34  09 
37  30 
40  49 
44  08 
47  28 
50  46 
54  04 
57  22 
60  39 

63  54 
67  10 
70  23 

INST.  AT  3.         s=  i°  oo'. 

INST.  AT  4.        j  —  1°  40'. 

No.  of 

Deflection  from 

Diff.  of 

No.  of 

Deflection  from 

Diff.  of 

Point. 

aux.  tan. 

Deflection. 

Point. 

aux.  tan. 

Deflection. 

0 

36'  40" 

Q'  10" 

O 

1°  O2'  30" 

10'  50" 

I 

27  30 

12   30 

I 

51   40 

14    10 

2 

3 

15 

00 

15 
2O 

2 

3 

37  30 

20 

17   30 
2O 

4 

20 

4 

00 

22    30 

25 

5 
6 

7 
8 

9 

10 

ii 

12 
13 
14 
15 

16 

42    30 
lp    08    20 

i     37  30 

2       10   OO 

2     45   50 
3     24  59 
4     07  28 
4     53   17 
5     42  25 
6     34   52 
7     30  37 
8     29  40 

25    50 

29  10 

32    30 

35   50 
39  09 
42  29 

45  49 
49  08 
52  27 

55  45 
59  °3 

62    21 

5 
6 

8 
9 

10 

ii 

12 
13 

M 
15 

16 

25 

52  30 

I      23    20 

i     57  30 
2     35  oo 
3     ID  50 
3     59  59 
4     47  28 
5     38  16 
6     32  24 
7     29  50 
8     30  34 

27   30 
30   50 
34  10 
37  30 
40  50 
44  09 
47  29 
50  48 
54  08 
57  26 
60  44 

64   02 

17 

18 

19 

20 

9     32  01 
10     37  37 
ii     46  29 
12     58  35 

65    36 

68  52 
72  06 

17 

18 

19 

20 

9     34  36 
10    41   55 
ii     52  29 
13     06  i  8 

67    19 

70  34 
73  49 

53 


TABLE  II. — DEFLECTION  ANGLES. 


INST.  AT  5.    s  =  2°  30'. 

INST.  AT  6.    j  =  3°  30'. 

No.  of 

Deflection  from 

Diff.  of 

No.  of   Deflection  from      Diff.  of 

Point. 

aux.  tan. 

Deflection. 

Point. 

aux.  tan.          Deflection. 

0 
I 

i°  35'  oo" 

I     22    30 

.    .12'  30" 

0 

I 

2°  14'  10" 
2    OO  OO 

14'  10" 

2 
3 
4 
5 

I     06    40 

47  30 

25 

00 

15    50 
19  10 

22    30 

25 

2 

3 
4 

5 

I    42    30 
I     21    40 

57  30 
3° 

17    30 
20    50 

24  10 

27    30 

6 

3° 

30 

6 

oo 

30 

7 
8 

9 

10 

ii 

12 
13 
14 
15 

16 

17 

18 

20 

1    02    30 
I     38    20 
2     17    30 

3   oo  oo 

3   45   50 
4  34  59 
5   27  28 
6   23  15 

7    22    23 

8    24  48 
9   30  31 
10  39  32 
ii    51  48 

13    07    20 

32    3O 

35  50 
39  10 
42  30 
45  50 
49  09 
52  29 
55  47 
59  08 
62  25 

65  43 
69  01 

72  16 

75  32 

7 
8 

9 

10 

ii 

12 

13 
14 

15 

16 

17 

18 

19 

20 

35 

I     12    30 

i    53  20 
2   37  30 
3   25  oo 
4   15  49 

5   09  58 
6   07  27 

7   08   15 

8     12    21 

9    19  46 
10  30  28 
ii    44  27 
13  oi  41 

35 
37  30  , 
40  50 
44  10 
47  30 
50  49 
54  09 
57  29 
60  48 
64  06 
67  25 
70  42 
73   59 
77  14 

INST.  AT  7.    ,$•  =  4°  40'. 

INST.  AT  8.    s  =  6°  co'. 

No.  of 

Deflection  from 

Diff.  of 

No.  of 

Deflection  from 

Diff.  of 

Point. 

aux.  tan. 

Deflection. 

Point. 

aux.  tan. 

Deflection. 

0 
I 
2 
3 
4 

6 

7 

3°  00'  00" 
2   44   10 

2     25    00 
2    O2    30 
I     36   40 
I    07    30 

35 

00 

*  5'  50" 
19  10 

22    30 
25    50 

29  10 

32    30 

35 

0 
f 
2 

3 
4 

5 
6 
7 

3°52/3i// 
•      3   35  oo 
3    14  10 

2     50   OO 
2     22    30 
I     51    40 
I     17    30 
40 

17'  31" 
20    50 

24  10 

27    30 
30    50 

34  10 
37  30 

8 

4° 

40 

8 

OO 

40 

9 

10 

it 

12 
13 
14 
15 

16 

17 

18 

I     22    30 

2    08    20 
2     57    30 

3   50  oo 
4  45  49 
5   44  58 
6   47  26 

7    53  14 
9   02  19 
10   14  43 

42  30 
45  50 
49  10 

52  30 
55  49 
59  °9 
62  28 
65  48 
69  05 
72  24 
75  4i 

9 

10 

ii 

12 
13 

14 
15 

16 

17 

18 

45 

i    32  30 

2     23    20 

3    17  30 
4   15  oo 
5    15  49 
6    19  58 
7   27  26 
8   38  13 
9   52  18 

45 
47  30 
50  50 
54  10 
57  30 
60  49 
64  09 
67  28 
70  47 
74  05 
77  22 

19 

20 

ii    30  24 
12   49  21 

78  57 

19 

20 

ii   09  40 

12    30    20 

80  40 
J 

54 


TABLE  II. — DEFLECTION  ANGLES. 


INST.  AT  g.    ^  =  7°  30'. 

INST.  AT  10.    j  =  9°  10'. 

No.  of 

Deflection  from 

Diff.  of 

No.  of  Deflection  from 

Diff.  of 

Point. 

aux.  tan.  , 

Deflection. 

Point.          aux.  tan. 

Deflection. 

0 
I 

2 

3 
4 
5 
6 

7 

8 
9 

4°  5i'  41" 
4   32  31 
4   10  oi 
3   44  10 
3    15  oo 
2   42  30 

2    06    40 
I     27    30 

45 

00 

19'  10" 

22    30 
25    51 

29  10 

32    30 

35  50 
39  10 
42  30 
45 

0 

I 
2 

3 
4 

6 

7 
8 

9 

5°57/32// 
5    36  42 
5    12   31 
4  45  oi 
4    14  10 
3   40  oo 
3   02  30 

2    21    40 

i    37  30 
50 

20'  50" 
24   II 
27   30 
30  51 
34  10 
37  30 
40  50 
44  10 
47  30 

10 

50 

50 

10 

CO 

5° 

ii 

12 
•    13 
14 
15 

16 

17 

18 
19 

20 

i   42  30 

2     38    20 

3    37  30 
4  40  oo 
5    45  49 
6   54  57 
8   07  25 
9   23  ii 
10  42  16 

12    04    38 

52  30 
55  50 
59  i° 
62  30 

65  49 
69  08 
72  28 
75  46 
79  °5 

82    22 

ii 

12 
13 

15 

16 

17 

18 

19 
20 

55 
i    52  30 
•2   53  20 
3   57  30 
5   05  oo 

6    15  49 
7   29  57 
8   47  24 
10  08  10 
ii    32  14 

55 
57  30 
60  50 
64  10 
67  30 
70  49 
74  08 
77  27 
80  46 
84  04 

INST.  AT  ii.    s  =•  11°  oo'.                         INST.  AT  12.    j  =  13°  oo'. 

No.  of 

Deflection  from 

Diff.  of 

No.  of 

Deflection  from 

Diff.  of 

Point. 

aux.  tan. 

Deflection. 

Point 

aux.  tan. 

Deflection. 

0 
I 
2 

3 
4 

7°  jo'  04" 
6   47  33 
6   21  42 
5    52  32 
5    20  oi 

22'  31" 

25    51 
29  10 

32  31 

1?    c  T 

O 
I 

2 

3 
4 

8°  29'  16" 
8   05  05 
7   37  34 
7   06  43 
6   32  32 

24'  Ii" 

27   31 
30   51 
34  ii 

5 
6 

4   44  10 
4  05  oo 

•   39  I0 

6 

5    55  oi 
5    T4  IT 

40  50 

8 
9 

10 

ii 

3     22    30 
2     36    40 

i    47  30 

55 

00 

42  3° 
45  50 
49  10 
52  30 
55 

7 
8 

9 

10 

ii 

4  30  oo 
3  42  30 
2   51  40 

i    57  30 

I    OO   OO 

44  ii 

47  30 
50  50 
54  10 

57  30 

Co 

60 

12 

I     OO    OO 

12 

^     oo 

62  30 

65    • 

13 
14 

15 

16 

17 

18 

20 

2    02    30 

3   08  20 
4   17  30 
5    29  59 
6   45  48 
8   04  57 
9   27  24 
10   53  09 

65   50 
69  10 
72  30 
75  49 
79  °9 
82  27 

85  45 

13 
14 

16 

17 

18 

19 

20 

i   05  oo 

2     12    30 

3   23  20 
4   37  30 
5    54  59 
7    15  48 
8    39  56 
10  07  23 

67  30 
70  50 
74  10 
77  29 
80  49 
84  08 
87  27 

55 


TABLE  II. — DEFLECTION  ANGLES. 


INST.  AT  13.      j=  15°  10'. 

INST.  AT  14.    j  =  17°  30'. 

No.  of  Deflection  from;      Diff.  of 

No.  of    Deflection  from      Diff.  of 

Point.          aux.  tan. 

Deflection. 

Point. 

aux.  tan.          Deflection. 

O 
I 
2 

9°  55'  10" 
9   29   18 
9   oo  06 

25'  52" 

29    12 

O 

I 
2 

n°  27'  45" 
ii   oo  13 
10  29  20 

27'  32" 
30  53 

3 
4 
5 
6 

7 
8 

9 

10 

8   27  35 
7   5i  44 
7   12  32 
6   30  02 

5   44  ii 
4   55  oo 
4  02  30 
3   06  40 

32    31 

35  5i 
39  12 
42  30 
45  5i 
49  ii 
52  30 
55  50 
59  1° 

3 
4 

5 
6 

7 
8 

9 

10 

9   55  08 
9    17  36 
8   36  45 
7   52  33 
7  05  02 
6    14  ii 
5    20  oo 

.4    22    30 

34  12 
37  32 
4°  5i 
44  12 
47  31 
50  51 
'   54  ii 
57  30 
60  50 

ii 

2  07  30 

62  ^o 

ii 

3   21  40  . 

64  10 

12 
13 

'or°  r  *: 

12 
13 

2     17    30 
1     IO   OO 

67  30 

70 

70 

14 

I     10   00              i       _ 

14 

00 

15 

16 

17 

18 

19 

20 

2    22    30 

3  38  20 
4  57  30 
6   19  59 

7  45  48 
9   14  56 

72  30 
75   50 
79  i° 
82  29 

85  49 
89  08 

15 

16 
17 
IB 

'9 

20 

i    15  oo 

2    32    30 

3   53  20 
5    17  30 
6  44  59 

8    15  48 

75 
77  30 
80  50 
84  10 
87  29 
90  49 

INST.  AT  15.    j  —  20°  co'. 

INST.  AT  16.    s  =  22°  40'. 

No.  of 
Point. 

Deflection  from 
aux.  tan. 

Diff.  of 
Deflection. 

No.  of 
Point. 

Deflection  from 
aux.  tan. 

Diff.  of 
Deflection. 

O 

I 
2 

3 
4 

6 

8 

1  3°  07'  03" 
12    37  49 
12   05   16 
ii    29  23 

10    50    TO 

10  07  37 
9   21  45 
8   32  34 
7  40  02 

29'  14" 

32  33 
35  53 
39  J3 
42  33 
45   52 
49  ii 
52  32 

O 
I 
2 

3 
4 
5 
6 

7 
8 

14°  53'  03" 

14    22    09 

13   47  54 

13     10    20 
12     29    26 

ii   45  12 
10   57  39 
10  06  46 
9    12  34 

30'  54" 
34  15 
37  34 
40  54 
44  14 
47  33 
50  53 
54  12 

9 

10 

ii 

12 
13 
14 

15 

6  44  ii 
5   45  oi 
4  42  30 
3   36  40 
2   37  30 
i    15  oo 
oo 

55  51 
59  10 
62  31 

65  50 
69  10 

72  30 

75 
So 

9 
10 
ii 

12      • 
13 
14 
15 

8    15  03 
6    14  ii 
6    10  qi 
5   02  30 
3    5i  40 
2   37  30 

I     2O  OO 

57  3T 
60  52 
64  10 

67  3i 
70  50 
74  10 

77  30 
80 

16 

I    20  OO 

16 

00 

17 
18 

19 

20 

2    42    30 

4  08  20 
5   37  30 
7  09  59 

85  50 
89  10 
92  29 

17 

18 

19 

20 

i   25  oo 

2    52    30 

4  23  20 
5   57  30 

87  30 
90  50 
94  10 

56 


TABLE    II. — DEFLECTION  ANGLES. 


INST.  AT  17.    j  =  25°  30'. 

INST.  AT  18,    -y  —  28°  30'. 

No.  of  Deflection  from!      DiiL  of 

No.  of 

Deflection  from 

Diff.  of 

Point. 

aux.  tan. 

Deflection. 

Point. 

aux.  tan. 

Deflection. 

0 

I 
2 

3 
4 
5 
6 

7 

8 

9 

10 

ii 

12 

16°  45'  48' 
16    13   ii 
15    37   15 
14    57  59 
14    15   24 
13    29  29 

12    40    14 

ii    47  41 
10   51  47 

9    52  35 
8    50  03 
'  7   44  12 
6   35  oi 

32'  37" 
36  56 
39  i6 
42  35 
45  55 
49  15 
52  33 
55  54 
59  12 
62  32 

65   51 

69    IT 

O 
I 

2 
3 
4 

5 
6 

7 

8 
9 

10 

ii 

12 

IS    10  59 
17    33  21 
16   52  23 
16   08  05 

15     20    28 

14  29  32 

13   35  17 
12    37  42 

ii    36  49 
10  32  36 
9   25  03 
8    14  12 

34'  1  8" 
37  38 
40  58 
44  18 
47  37 
50  56 
54  15 
57  35 
60  53 
64  13 
67  33 
70  51 

13 

5     22    30 

72    31 

13 

7  oo  oi 

74  it 

14 

15 

16 

17 

4   06  40 
2   47  30 
i   25  oo 
oo 

75  5° 
79  10 
82  30 

85 

14 

15 

16 
17 

5   42  30 
4   21  40 
2    57  30 
i   30  oo 

77  3r 
80  50 
84  10 
87  30. 

18 

i   30  oo 

9° 

18 

00 

9° 

19 

20 

3   02  30 
4  38  20 

92  30 
95  50 

19 

20 

i    35  oo 
3   12  30 

95 

97  30 

INST.  AT  19.    j  =  31°  40'. 

INST.  AT  20.     j-  =  35°  oo'  . 

No.  of   Deflection  from 

Diff.  of 

!  No.  of 

Deflection  from 

Diff.  of 

Point. 

aux.  tan. 

Deflection  . 

i  Point. 

aux.  tan. 

Deflection. 

0 
I 

2 

3 
4 

5 
6 

7 
8 

9 

10 

n 

12 
13 

20°  5  1'  33" 
20    15    32 
19    36    II 

18    53  3i 
18   07  31 

17     IS     12 

16   25   33 
15    29  36 
14   30  20 
13    27  44 

12     21    50 
II     12    36 

10  oo  04 
8   44  12 

36'  01  " 
39  21 
42  40 
46  oo 
49  J9 
52  39 
55  57 
59  16 
62  36 

65  54 
69  14 

75  32 
75   52 
70    II 

0 
I 
2 

3 
4 
5 
6 

8 
9 

10 

ii 

12 
13 

23°  04'  36" 
22    26    52 

21   45  48 

21    01    25 
20     13    42 
19    22    40 

18   28   19 

17   30  39 
16   29  40 
15    25   23 
14    17  46 
13   06  51 
ii    52  37 
10   35  04 

37'  44" 
41  04 
44  23 
47  43 
51  02 
54  21 
57  40 
60  59 
64  17 
67  37 
7'^>  55 
74  -14 
77  33 
80  52 

M 
15 

16 

17 

18 

19 

7    25   oi 
6  02  30 
4   36  40 

3   07  30 

i    35 
oo 

82   31 
85.50 
89  10 
92  30 
95 

IOO 

14 
15 

16 

17 

18. 

19 

9   r4  12 
7   50  oi 

6    22    30 

4   51  40 
3    17  30 
i   40 

84  ii 

87  31 
90  50 
94  10 
97  30 
1  60 

20 

i    40 

20 

00 

57 


TABLE    III. 


DEGREE  OF  CURVE  AND  VALUES  OF  THE  COORDINATES  x  AND 
y,  FOR  EACH  CHORD-POINT  OF  THE  SPIRAL  FOR  VARIOUS 
LENGTHS  OF  THE  CHORD. 


f.  CHORD-LENGTH  =  IO. 

n. 

nc. 

/>. 

y> 

Xt 

Log  x. 

j 

10 

i°  40'  oo" 

IO.OOO 

0.0145 

8.162696 

2 

20 

3   20  02 

20.  ooo 

.0727 

8.861664 

3 

30 

5  oo  06 

29.99-) 

.  2036 

9.308815 

4 

40 

6  40  13 

39.996 

.4363 

9.639792 

5 

50 

8   20  26 

49.990 

.7998 

9.903002 

6 

60 

10  oo  45 

59-976 

1.323 

0.121624 

7 

70 

II   41  12 

69.951 

2.035 

0.308622 

8 

80 

13   21  48 

79.907 

2.965 

0.471991 

9 

90 

ID  02  34 

89.838 

4.140 

0.617015 

10 

100 

16  43  3i 

99.732 

5-589 

0.747370 

ii 

no 

18  24  42 

109  578 

7-340 

0.805712 

12 

120 

20   06  07 

119-359 

9.419 

0.974022 

13 

130 

21  47  48 

129.059 

11.853 

1.073818 

14 

140 

23  29  46 

138.655 

14.665 

1.166281 

15 

150 

25   12  02 

148.125 

17.879 

.252352 

16 

160 

26  54  39 

157.439 

21.517 

.332788 

17 

170 

28  37  38 

166.569 

25-598 

.408205 

18 

1  80 

3O   21  01 

175-479 

30  138 

.479112 

19 

i  go 

32   04  48 

184.132 

35.150 

•545931 

20 

200 

33  49  02 

192.487 

40.645 

.609013 

35  33  46 

TABLE    III. 


r.  CHORD-LENGTH  =  n. 

„  f  »c. 

Ds. 

y- 

x. 

Log  x. 

I 

II 

i°  30'  55" 

1  1  .  CO3 

0.0160 

8.204089 

2  j   22 

3  oi  50 

22.OOO 

.0800 

8.903057 

3 

33 

4  32  48 

32.999 

.2240 

9.350208 

4 

44 

6  03  48 

43.996 

•4799 

9  681185 

5 

55 

7  34  52 

54.989 

.8798 

9.944394 

6 

66 

9  06  01 

65.974 

1.456 

0.163017 

7 

77 

10  37  16 

76.946 

2.239 

0.350015 

8 

88 

12  08  37 

87.898 

3.261 

0-513384 

9 

99 

13  40  06 

98.822 

4-554 

0.658408 

JO 

no 

15  ii  44 

109.706 

6.148 

0.788763 

IT 

121 

16  43  3i 

120.536 

8.074 

0.907104 

12 

132 

18  15,29 

I3L295 

10.361 

1.015415 

13 

143  !  19  47  39 

141.965 

13-038 

1.115210 

14 

154  I  21  20  01 

152.521 

16.131 

1.207674 

15 

165    22  52  38 

162.937 

19.667 

.293745 

16  |  176 

24  25  29 

173.183 

23.669 

.374180 

17 

I87 

25  58  36 

183.226 

28.158 

.449598 

18 

198 

27  32  01 

193.027 

33.152 

.520505 

19 

20g 

29  05  45 

202.545 

38.665 

•587323 

20 

220 

30  39  48 

2H.735 

44.710 

.650405 

32  14  ii 

.  c.  CHORD-LENGTH  ±=  12. 

;/. 

11C. 

Ds.   - 

y- 

jr. 

Log  jr. 

I 

12 

i°  23'  20" 

12.000 

0.0175 

8.241877 

2 

24 

2  46  41 

24.OOO 

.0873 

8.940845 

3 

36 

4  10  03 

35-999 

•2443 

9o87997 

4 

48 

5  33  28 

47.996 

.5236 

9.718974 

5 

60 

6  56  55 

59-988 

.9598 

9.982183 

6 

72 

8  20  26 

71.971 

1.588 

o.  200806 

7 

84 

9  44  oi 

83.941 

2.442 

0.387803 

8 

96 

ii  07  42 

95.889 

3.558 

0.551172 

9 

108 

T2  31  28 

107.806 

4.968 

0.696196 

10 

120   13  55  21 

119.679 

6.707 

0.826551 

ii 

132 

15   19  22 

I3L493 

8.8c8 

0.944893 

12 

144 

16  43  31 

143.231 

11-303 

.053204 

13 

156 

18  07  48 

154.871 

14.223 

.152999 

14 

168   19  32  15 

166.386 

17-598 

.245462 

15 

180   20  56  53 

177-749 

21-455 

.331533 

16   192   22  21  43 

188.927 

25.821 

.411969 

17  204 

23  46  44 

199.883 

30.718 

.487386 

18   216   25  ii  59 

210.575 

36.165 

•558293 

19   228   26  37  28 

220.958 

42.181 

.625113 

20 

240 

28  03  12 

230.984 

48.774 

.688194 

29  29  12 

59 


TABLE    III. 


c.  CHORD-LENGTH  =  13. 

n. 

nc. 

Ds. 

y- 

X. 

Log  x. 

l 

13 

i°  16'  55" 

13.000 

0.0189 

8.276639 

2 

26 

2  33  52 

26.000 

•0945 

8.975607 

3 

39 

3  50  49 

38.999 

.2647 

9  422759 

4 

52 

5  07  48 

51-995 

•5672 

9.753736 

5 

65 

6  24  49 

64.987 

1.040 

0.016945 

6 

78 

7  4i  53 

77.969 

1.720 

0.235568 

7 

91 

8  59  oo 

90.936 

2.646 

0.422565 

8 

104 

10  16  12 

103.879 

3.854 

0.585934 

9 

117 

ii  33  28 

116.789 

5-382 

0.730959 

JO 

130 

12  50  49 

129.652  - 

7.266 

0.861313 

ii 

143  !  14  08  16 

142.451 

9-542 

0-979655 

12 

156 

15  25  50 

155.167 

12.245 

.087966 

13 

169 

16  43  30 

167.776 

15.409 

.187761 

14 

182 

i3  01  18 

180.252 

19.064 

.280224 

15 

195 

IQ  19  14 

192.562 

23.243 

.366295 

16 

208 

20  37  20 

204.671 

27.972 

.446731 

17 

221 

21  55  34 

216.540 

33.277 

.522148 

18 

234 

23  14  oo 

228.123 

39-179 

.593055 

19 

247 

24  32  35 

239.371 

45.696 

.659874 

20 

260 

25  5i  23 

250.233 

52.839 

1.722956. 

27  10  23 

c.  CHORD-LENGTH  =  14. 

71. 

11C. 

A 

y- 

X. 

Log  x. 

I 

14 

i°  n'  26" 

14.000 

0.0204 

8.308824 

2 

28 

2  22  52 

28.  coo 

.1018 

9.007792 

3 

42 

3  34  19 

41.999 

.2851 

9-454943 

4 

56 

4  45  48 

55.995 

.6108 

9.785920 

5 

70 

5  57  18 

69.986 

I.I2O 

0.049130 

6 

84 

7  "8  51 

83.966 

1.852 

0.267752 

7 

98 

8  20  26 

97.931 

2.849 

0.454750 

8 

112 

9  32  04 

111.870 

4-I5I 

0.618119 

9 

126 

10  43  47 

125  773 

5.796 

0.763M3 

10 

140 

ii  55  33 

139.625 

7.825 

0.893498 

TI 

154 

13  07  24 

153-409 

10.276 

1.011840 

12 

1.68 

14  19  20 

167.103 

13.187 

1.120150 

13 

182 

15  31  22 

180.682 

16.594 

1.219946 

14 

196 

16  43  29 

194.117 

20.531 

1.312409 

15 

2IO 

*7  55  44 

207.374 

25.031 

1,398480 

16 

224 

19  c6  05 

220.415 

30.124 

1.478915 

17 

238  i  20  20  34 

233.196 

35-837 

1-554333 

1-8 

252   21  33  ii 

-245-670 

42.193 

1.625240 

J9 

266   22  45  56 

257./35 

49.211 

1.692059 

20 

280 

23  53  51 

269.481 

56.903 

I.755I4I 

25  ii  55 

60 


TABLE   III. 


c.  CHORD-LENGTH  =  15. 

?/.  j  nc. 

Ds. 

y. 

x. 

Log  x. 

i    IS 

i°  06'  40" 

15.000 

0.0218 

8.338787 

2 

30 

2  13  2O 

30.  ooo 

.1091 

9-037755 

3- 

45 

3  20  02 

44.998 

.3054 

9.484907 

4 

60 

4  26  44 

59.994 

.6545 

9.815884 

5 

75 

5  33  28 

74.984 

1.200 

0.079093 

6 

90 

6  40  13 

89.964 

1.985 

0.297716 

7 

105 

7  47  oi 

104.926 

3-053 

0484713 

8 

1  20 

8  53  5i 

119.861 

4-447 

0.648082 

9 

135 

10  oo  45 

134.757 

6.2IG 

0.793107 

10 

150 

ii  07  41 

149-599 

8.384 

o  923461 

ii 

165 

12  14  41 

164.367 

II.OIO 

1.041803 

12 

1  80 

13  21  47 

179.039 

14.129 

1.150114 

13 

195 

14  28  56 

193.588 

17-779 

.249909 

14 

210 

15  36  09 

207.983 

21.997 

.342372 

15 

225 

16  43  28 

222.187 

26.819 

•428443 

16 

240 

17  50  54 

236.159 

32.276 

.508879 

17 

255 

18  58  25 

249.853 

38.397 

.584296 

18 

270 

20  06  02 

263  218 

45.207 

.655203 

19 

285 

21  13  47 

276.198 

52.726 

1.722022 

20 

300 

22  21  39 

288.730 

60.968 

1.785104 

23  29  48 

c.  CHORD-LENGTH  =  16. 

;/. 

nc 

Ds.  - 

y- 

jr. 

Log  jr. 

I 

16 

i°  02'  30" 

1  6  ooo 

0.0233 

8.366816 

2 

32 

2  05  00 

32.000 

.1164 

9.065784 

3 

48 

3  07  31 

47.998 

.3258 

9-5I2935 

4 

64 

4  10  03 

63.994 

.6981 

9.843912 

5 

80 

5  12  36 

79-983 

1.260 

0.107122 

6 

96 

6  15  ii 

95.961 

2.117 

0.325744 

7 

112 

7  17  47 

111.921 

3-256 

0.512742 

8 

128 

8  20  26 

127.852 

4-744 

0.676111 

9 

144 

9  23  07 

143-74I 

6.  624 

o  821135 

10 

1  60 

10  25  51 

159-572 

8-943 

0.951490 

ii 

I76 

ii  28  37 

175.325 

11.744 

.069832 

12 

IQ2 

12  31  28 

190.975 

15.071 

.178142 

13 

208 

13  34  21 

206.494 

18.964 

.277938 

14 

224 

14  37  20 

221.848 

23.464 

.370401 

15 

240 

15  4°  21 

236.999 

28.607 

.456472 

16 

256 

16  43  28 

251.903 

34.428 

•53f>9°7 

17 

272 

17  46  40 

266.510 

40.957 

.612325 

18 

288 

18  49  57 

280.766 

48.221 

.683232 

19 

304 

19  53  20 

294.611 

56.241 

.750051 

20 

320 

20  56  49 

307.979 

65.032 

•813133 

22  00  23 

61 


TABLE    III. 


t.  CHORD-LENGTH  =  17. 

//. 

11C. 

£>*. 

y+ 

-V. 

Log  x. 

i 

17 

o°  58'  49" 

17.000 

0.0247 

8.393M5 

2 

34 

i  57  33 

34.000 

.1236 

9.092113 

3 

51 

2  56  27 

50.998 

.3461 

9  539264 

4 

68 

3  55  19 

67.994 

•7417 

9.870241 

5 

85 

4  54  12 

84.982 

1.560 

O.I3345I 

6 

IO2 

5  53  06 

101.959 

2.249 

0.352073 

7 

II9 

6  52  oo 

118.916 

3-460 

0.539071 

8 

I36 

7  50  57 

1  3  5-  -842 

5.040 

o.  702440 

9 

153 

8  49  55 

152.725 

7.038 

0.847464 

10 

I70 

9  48  56 

169.545 

9.502 

0.977819 

ii 

I87 

10  48  oo 

186.282 

12.478 

1.096161 

12 

204 

ii  47  07 

202.911 

16.013 

1.204471 

13 

221 

12  46  15 

219.400 

20.150 

1.304267 

14 

238 

13  45  27 

235.7H 

24-930 

•396730 

15 

255 

14  44  44 

251.812 

30.395 

.482801 

16 

272 

15  44  03 

267.647 

36.579 

.563236 

17 

289 

16  43  27 

283.167 

43.5I6 

.638654 

18 

306 

17  42  56 

298.314 

5L234 

.709561 

19 

323 

18  42  29 

313.024 

59.756 

.776380 

20 

340 

19  42  07 

327.228 

69.097 

.839462 

20  41  49 

f.  CHORD-LENGTH  =  18. 

;/. 

ftf. 

$« 

y. 

a~. 

Log^r. 

I 

18 

o°  55'  33" 

18.000 

0.0262 

8.417968 

2 

36 

i  51  07 

36.000 

.1309 

9.116937 

3 

54 

2  46  40 

53.998 

.3665 

9.564088 

4 

72 

3  42  16 

71.993 

•.7853 

9.895065 

5 

90 

4  37  5i 

89.981 

1.440 

0.158274 

6 

1  08 

5  33  28 

107-957 

2.382 

0.376897 

7 

126 

6  29  05 

125.911 

3-663 

0.563894 

8 

144 

7  24  45 

143.833 

5-337 

0.727263 

9 

162 

8  20  26 

161.708 

7-452 

0.872288 

10 

180 

9  16  08 

179.518 

10.  06  1 

1.002643 

ii 

198 

10  ii  54 

197.240 

13.212 

1.120984 

12 

216 

ii  07  41 

•214.847 

16.955 

1.229295 

13 

234 

12  03  31 

232.  3c6 

21-335 

1.329090 

14 

252 

12  59  24 

249.579 

26.397 

I.42I554 

15 

270 

13  55  20 

266.624 

32.183 

1.507624 

16 

288 

14  51  18 

283.391 

38.731 

1.588060 

17 

306 

15  47  20 

299.824 

46.076 

1.663477 

18 

324 

16  43  27 

315.862 

54.248     1.734385 

*9 

342 

17  39  37 

33L437 

63.271     1.801203 

20 

360 

18  35  5i 

346.476 

73.161 

1.864285 

19  32  08 

62 


TABLE    III. 


c.  CHORD-LENGTH  =  19. 

n. 

lie. 

Ds. 

y> 

x. 

Log^r. 

i 

19 

o°  52'  38" 

19.000 

0.0276 

8.441450 

2 

3^ 

i  45  16 

38.000 

.1382 

9.140418 

3 

57 

2  37  54 

56.998 

.3869 

9-587569 

4 

76 

3  30  34 

75-993 

.8290 

9-9i8546 

5 

95 

4  23  13 

94.980 

1.520 

0.181755 

6 

H4 

5  15  54 

113-954 

2.514 

0.400378 

7 

133 

6  08  36 

132.906 

3.867 

0.587376 

8 

152 

7  01  19 

151.824 

5.633 

0.750744 

9 

171 

7  54  03 

170.692 

7.866 

0.895769 

10 

190 

8  46  49 

189.491 

10.620 

1.026124 

ii 

209 

9  39.  36 

208.198 

13.947 

1.144465 

12 

228 

10  32  26 

226.783 

17.897 

1.252776 

13 

247 

ii  25  18 

245.212 

22.520 

.352571 

14 

266 

12  18  12 

263.445 

27.863 

•445035 

15 

285 

13  II  09 

281.437 

33-971 

•53H05 

16 

304 

14  04  09 

299.135 

40.883 

.611541 

17 

323 

14  57  ii 

316.481 

48.636 

.686958 

18 

342 

15  50  16 

333-410 

57.262 

.757866 

19 

361 

16  43  25 

349-85i 

66.786 

.824684 

20 

380 

17  36  33 

365-725 

77.226 

.887766 

18  29  54 

•  c.  CHORD-LENGTH  =  20. 

11. 

nc. 

Ds.   * 

}'- 

X. 

Log  x. 

i 

20 

o°  50'  oo" 

20.000 

0.0291 

8.463726 

2 

40 

i  40  oo 

40.000 

.1454 

9.162694 

3 

60 

2  30  01 

59.998 

.4072 

9.609845 

4 

80 

3  20  02 

79-993 

.8726 

9.940822 

5 

IOO 

4  10  03 

99-979 

i.  600 

0.204032 

6 

120 

5  oo  05 

119-952 

2.646 

0.422654 

7 

140 

5  5°  °8 

139.901 

4.071 

0.609652 

8 

1  60 

6  40  13 

159  8i5 

5.930 

0.773021 

9 

180 

7  30  18 

179.676 

8.280 

0.918045 

10 

200 

8  20  26 

199.465 

11.179 

1.048400 

ii 

22O 

9  i°  34 

219.156 

14.681 

1.166742 

12 

240 

10  oo  44 

238.719 

18.839 

1.275052 

13 

260 

10  50  56 

258.118 

23-705 

1.374848 

14  1  280 

ii  41  10 

277-310 

29.330 

1.467311 

15 

3OO 

12  31  26 

296.249 

35-759 

1.553382 

16 

320 

13  21  45 

314.879 

43-035 

1.633817 

17 

340 

14  12  c6 

333.138 

51.196 

1.709235 

18 

360 

15  02  29 

350.958 

60.276 

1.780142 

19 

380 

15  52  55 

368.264 

70.301 

1.846961 

20 

400 

16  43  25 

384.974 

81.290 

1.910043 

17  33  & 

TABLE    III. 


c.  CHORD-LENGTH  =  21. 

i 

n. 

nc. 

Ds. 

}'• 

X. 

Log.  x. 

I 

21 

o°  47'  37" 

2  1  .  OOO 

0.0305 

8.484915 

2 

42 

i  35  14 

42.000 

.1527 

9.183883 

3 

63 

2  22  52 

62.998 

.4276 

9-63I035 

4 

84 

3  10  30 

83.992 

.9162 

9.962012 

5 

105 

3  58  08 

104.978 

1.680 

0.225221 

6 

126 

4  45  47- 

125.949 

2.779 

0.443844 

7 

147 

5  33  27 

146.  896 

4.274 

0.630841 

8 

168 

6  21  08 

167.805 

6.226 

0.794210 

9 

189 

7  08  50 

188.660 

8.694 

0.939235 

TO 

2IO 

7  56  33 

209.438 

11.738 

.069589 

II 

231 

8  44  18 

230.114 

15.415 

.187931 

12 

252 

9  32  03 

250.655 

19.781 

.296242 

13 

273 

10  19  51 

271.023 

24.891 

•396037 

M 

294 

n  07  40 

291.176 

30.796 

.488500 

15 

315 

ii  55  3i 

3II.062 

37.547 

.574571 

16 

336 

12  43  24 

330.623 

45.186 

.655007 

17 

357 

13  31  20 

349-795 

53.756 

.730424 

18 

378 

14  19  17 

368.  506 

63.289 

.801331 

IQ 

399 

15  07  17 

386.677 

73.816 

.868150 

15  55  19 

r.  CHORD-LENGTH  =  22. 

;/. 

nc. 

Ds. 

y- 

X. 

Log.  x. 

i 

22 

45'  27" 

22.000 

0.0320  . 

8.505119 

2 

44 

i°  30  53 

44.  ooo 

.1600 

9  204087 

3 

66 

2  l6  22 

65.998 

.4480 

9.651238 

4 

88 

3  oi  50 

87.992 

•  9599 

9.982215 

5 

no 

3  47  18 

109.977 

1.760 

0.245424 

6 

132 

4  32  48 

I3L947 

2.911 

0.464047 

/ 

154 

5  18  18 

153.891 

4.478 

0.651045 

8 

176 

6  03  48 

I75-796 

6.522 

0.814414 

9 

198 

6  49  19 

197.643 

9.108 

0.959438 

10 

220 

7  34  5i 

219.411 

12.297 

.089793 

ii 

242 

8  20  25 

241.071 

16.149 

.208134 

12 

264 

9  06  oo 

262.591 

20.7^3 

.316445 

13 

286 

9  5i  36 

283.929 

26.076 

.416240 

14 

308 

10  37  13 

305042 

32.263 

.508704 

15 

330 

ii  22  53 

325.874 

39-335 

•594775 

1.6 

352 

12  08  34 

346.367 

47.338 

.675210 

17 

374 

12  54  16 

366.451 

56.3*5 

1.750628 

18 

396 

13  40  oi 

386.054 

66.303 

1.821535 

14  25  49 

0-!- 


TABLE    III. 


r.  CHORD-LENGTH  =  23. 

n. 

nc. 

D*. 

y- 

X. 

Log.  x. 

I 

23 

o'"'  43'  29" 

23.000 

0.0335 

8.524424 

2 

46 

I  26  58 

46.000 

.1673 

9.223392 

3 

69 

2  10  26 

68.998 

.4683 

9-670543 

4 

92 

2  53  56 

91.991 

1.004 

0.001520 

5 

H5 

3  37  26 

114.976 

.1.840 

0.264729 

6 

138 

4  20  56 

137-945 

3.043 

0.483352 

7 

161 

5  04  26 

160.886 

4.681 

0.670350 

8 

184 

5  47  58 

.  183.787 

6.819 

0.833719 

9 

207 

6  31  30 

206.627 

9.522 

0.978743 

10 

230 

7  15  04 

229.384 

12.856 

1.109098 

ii 

253 

7  58  38 

252.029 

16.883 

.227439 

12 

276 

8  42  13 

274  527 

21.665 

.335750 

*3 

299 

9  25  49 

296.835 

27.261 

•435545 

14 

322 

10  09  27 

318.907 

33.729 

.528009 

15 

345 

10  53  06 

340.686 

41.123 

.614080 

16 

368 

1  1  36  47 

362.110 

49.490 

•694515 

17 

391 

12  20  29 

383.108 

58.875 

.769933 

13  04  13 

r.  CHQRD-LENGTH  =  24. 

72. 

nc. 

Ds. 

;'• 

X. 

Log.  «r. 

I 

24 

41'  40" 

24.  ooo 

0.0349 

8.542907 

2 

48 

1°  23  20 

48.000 

.1745 

9.241875 

3 

72 

2  05  OO 

71.998 

.4887 

9.689027 

4 

96 

2  46  41 

95.991 

1.047 

0.020004 

5 

1  20 

3  28  22 

119-975 

1.920 

0.283213 

6 

144 

4  10  03 

143.942 

3.176 

0.501836 

7 

1  68 

4  5r  45 

167.881 

4.885 

0.688833 

8 

192 

5  33  28 

191-777 

7.115 

0.852202 

9 

216 

6  15  10 

215.611 

9.936 

0.997226 

10 

240 

6  5^  54 

239-358 

13.415 

1.127581 

n 

264 

7  38  39 

262.987 

17.617 

1.245923 

12 

288 

8  20  25 

286.463 

22.607 

i  354234 

13 
14 

312 
336 

9  02  12 
9  44  oo 

309.741 
332.773 

28.446 
35.196 

.1.454029 
1.546492 

15 

360 

10  25  48 

355.499 

42.910 

1.632563 

16 

384 

Ji  07  30 

377.854 

51.641 

1.712999 

17 

408 

ii  49  31 

399-765 

61.435 

1.788416 

12  31  25 

65 


TABLE   III. 


C.   L.tlU 

IvU-I^l^INLrl 

n  =  25. 

n. 

nc. 

Ds. 

y* 

X. 

Log.  x. 

I 

25 

o°  40'  oo" 

25.000 

0.0364 

8.560636 

2 

50 

I  2O  CO 

50.000 

.1818 

9.259604 

3 

75 

2  00  00 

74-997 

.5090 

9-706755 

4 

100 

2  40  OI 

99.991 

1.091 

0.037732 

5 

I25 

3  20  02 

124.974 

2.OOO 

0.300942 

6 

150 

4  oo  03 

149.940 

3-308 

0.519564 

7 

175 

4  40  04 

174.876- 

5.088 

0.706562 

8 

2GO 

5  20  06 

199.768 

7.412 

0.869931 

9 

225 

6  oo  09 

224-595 

IO.35O 

•014955 

10 

250 

6  40  13 

249-33I 

13-974 

.145310 

ii 

275 

7  20  17 

273-945 

18.351 

.263652 

12 

300 

8  00  22 

298.398 

23.548 

.371962 

13 

325 

8  40  28 

322.647 

29.632 

.471758 

14 

350 

9  20  35 

346.638 

36.662 

.564221 

'15 

375 

10  oo  43 

37°-3i  i 

44-698 

.650292 

16 

400 

10  40  52 

393-  598 

53-793 

.730727 

II  21  03 

f.     CHORD-LENGTH  =  26. 


n. 

nc. 

Ds. 

>'• 

X. 

Log.  x. 

I 

26 

o°  38'  28" 

26.000 

0.0378 

8.577669 

2 

52 

i  16  56 

52.000 

.1891 

9.276637 

3 

78 

i  55  24 

77.997 

.5294 

9.723789 

4 

104 

2  33  52 

103.990 

1.  134 

0.054766 

5 

130 

3  12  20 

129.973 

2.080 

0.317975 

6 

156 

3  50  48 

155.937 

3-440 

0.536598 

7 

182 

4  29  18 

181.871 

5.292 

0.723595 

8 

208 

5  07  48 

207.759 

7.708 

0.886964 

9 

234 

5  46  18 

233.579 

10.764 

1.031989 

10 

260 

6  24  48 

259-304 

14.533 

1.162343 

ii 

286 

7«03  20 

284.903 

19.085 

1.280685 

12 

312 

7  41  52 

310.334 

24.490 

1.388996 

13 

338 

8  20  25 

335-553 

30.817 

1.488791 

14 

364 

8  58  59 

360.504 

38.129 

1.581254 

15 

39° 

9  37  33 

385.124 

46.486 

1.667325 

10  1  6  OQ 

66 


TABLE    III. 


c.  CHORD-LENGTH  =  27. 

n. 

11C. 

Dt. 

}'• 

X. 

Log.  x. 

I 

27 

o°  37'  02" 

27.000 

0.0393 

8.594060 

2 

54 

i  14  04 

54.000 

.1963 

9.293028 

3 

81 

i  5i  07 

80.997 

.5498 

9.740179 

4 

108 

2  28  10 

107.990 

1.178 

0.071156 

5 

135 

3  05  12 

I34-972 

2.160 

0.334365 

6 

162 

3  42  15 

161.935 

3-573 

0.552988 

7 

189 

4  19  19 

188.866 

5-495 

0.739986 

8 

216 

4  56  23 

215-750 

8.005 

0.903355 

9 

243 

5  33  28 

242.562 

11.178 

1.048379 

10 

270 

6  10  32 

269.277 

15.092 

LI78734 

n 

297 

6  47  38 

295.860 

19.819 

1.297075 

12 

324 

7  24  44 

322.270 

25.432 

1.405386 

13 

35i 

8  or  51 

348.459 

32.002 

1.505181 

14   378 

8  38  59 

374-369 

39-595 

1.597645 

15 

405 

9  16  07 

399.936 

48.274 

1.683716 

9  53  16 

c.  CHORD-LENGTH  =  28. 

n. 

nc. 

/& 

y- 

jr. 

Log.  jr. 

I 

28 

o°  35'  42" 

28.000 

0.0407 

8.609854 

2 

56 

i  ii  26 

55-999 

.2036 

9.308822 

3 

84 

i  47  08 

83.997 

.5701 

9-755973 

4 

112 

2  22  52 

in.  990 

1.222 

0.086950 

5 

140 

2  58  36 

139-97I 

2.24O 

0.350160 

6 

168 

3  34  19 

1  67.933 

3-705   ' 

0.568782 

7 

196 

4  10  03 

195.862 

5.699 

o.755/So 

8 

224 

4  45  48 

223.740 

8.301 

0.919149 

9 

252 

5  21  32 

251.546 

H.592 

1.064173 

10 

280 

5  57  17 

279.251 

15.650 

1.194528 

ii 

308 

6  33  03 

306.818 

20.553 

1.312870 

12 

336 

7  08  50 

334.206 

26.374 

1.421180 

13 

364 

7  44  36 

361.365 

33.188 

1.520976 

14 

392 

8  20  24 

388.235 

4I.O62 

I.6I3439 

8  56  13 

67 


TABLE    III. 


CHORD-LENGTH  -  29. 


n. 

nc. 

Ds. 

y- 

X. 

Log.  oc. 

I 

29 

o°  34'  29" 

29.000 

0.0422 

8.625094 

2 

53 

i  08  58 

57-999 

.2109 

9.324062 

3 

8? 

I  43  27 

86.997 

•59°5 

9.771213 

4 

116 

2  17  56 

115-989 

1.265 

0.102190 

5 

145 

2  52  26 

144.970 

2.320 

0.365400 

6 

174 

3  26  55 

173-930 

3-837 

0.584022 

7 

203 

4  01  26 

202.857 

5.902 

0.771020 

8 

232 

4  35  56 

231.731 

8.598 

0.934389 

9 

261 

5  10  26. 

260.530 

12.006 

.079413 

10 

290 

5  44  57 

289.224 

16.209 

.209768 

ii 

319 

6  19  29 

317.776 

21.287 

.328110 

12 

348 

6  54  01 

346.142 

27-316 

.436420 

13 

377 

7  28  34 

374.271 

34-373 

.536216 

14 

406 

8  03  07 

402.100 

42.528 

.628679 

8  37  40 

CHORD-LENGTH  =  30. 


n. 

nc. 

Ds. 

y- 

jr. 

Log.  x. 

T 

3° 

o°  33'  20" 

30.000 

0.0436 

8.639817 

2 

60 

i  06  40 

59-999 

.2182 

9-338785 

3 

90 

i  40  oo 

89.997 

.6108 

9-785937 

4 

120 

2  13  20 

119.989 

1.309 

0.116914 

5 

150 

2  46  41 

149.969 

2.400 

0.380123 

6 

1  80 

3  20  02 

179.928 

3-970 

0.598746 

7 

210 

3  53  22 

209.852 

6.106 

0.785743 

8 

240 

4  26  44. 

239.722  . 

8.894 

0.949112 

9 

270 

5  oo  05 

269.514 

12.420 

.094137 

10 

3CO 

5  33  27 

299.197 

16.768 

.224491 

ii 

330 

6  06  49 

328.734 

22.021 

.^42833 

12 

360 

6  40  12 

358.078 

28.258 

.451144 

13 

39° 

7  13  36 

387.176 

35.558 

.550939 

7  47  oo 

68 


TABLE    ,„./ 


' 

/ 

c.  CHORD-LENGTH  =  31. 

n. 

nc. 

E, 

y 

*. 

Log  x. 

i 

31 

o°  32'  15" 

31.000 

0.0451 

8.654058 

2 

62 

i  04  31 

61.999 

.2254 

9.353026 

3 

93 

i  36  47 

92.997 

.6312 

9.800177 

4 

124 

2  09  C2 

123.988 

1-353 

0.131154 

5 

155 

2  41  18 

154.968 

2-479 

0.394363 

6 

1  86 

3  13  34 

185  925 

4.102 

0.612986 

7 

217 

3  45  50 

216.847 

6.309 

0.799984 

8 

248 

4  1  8  07 

247-713 

9.191 

0-963353 

9 

279 

4  50  24 

278.498 

12.834 

1.108377 

10 

310 

5  22  41 

309.  1  70 

17.327 

1.238732 

ii 

341 

5  54  59 

339.692 

22.755 

1.357073 

12 

372 

6  27  17 

370.014 

29.200 

1.465384 

13 

6  59  35 

400.082 

36.743 

1.565179 

7  3i  53 

\ 

CHORD-LENGTH  =  32. 

n. 

nc. 

Ds. 

y- 

X. 

Log*. 

I 

32 

o°  31'  15" 

32.000 

0.0465 

8.667846 

2 

64 

I  02  30 

63  999 

.2327 

9.366814 

3 

96 

i  33  45 

95-997 

.6516 

9.813965 

4 

128 

2  05  00 

127.988 

1.396 

0.144942 

5 

1  60 

2  36  16 

159.967 

2-559 

0.408152 

6 

192 

3  07  31 

101.923 

4-234 

0.626774 

7 

224 

3  38  47 

223.842 

6.513 

0.813772 

8 

256 

4  10  03 

255.703 

9.487 

0.977141 

9 

288 

4  4i  19 

287.481 

13.248 

1.122165 

10 

320 

5  12  36 

319.144 

17.886 

1.252520 

ii 

352 

5  43  53 

350.649 

23.489 

1.370802 

12 

384 

6  15  10 

381.950 

30.  142 

I.479r72 

13 

416 

6  46  28 

412.988 

37.929 

1.578968 

7  17  46 

69 


TABLE    III. 


c.  CHORD-LENGTH  =  33. 

n. 

12C. 

Ds. 

y- 

X. 

Log.  x. 

i 

33 

o°  30'  19" 

33.000 

0.0480 

8.681210 

2 

66 

I  OO  36 

65.999 

.2400 

9.380178 

3 

99 

i  30  55 

98.997 

.6719 

9.827329 

4 

132 

2  OI  13 

131.988 

1.440 

0.158306 

5 

165 

2  3I  32 

164.966 

2.639 

0.421516 

6 

198 

3  oi  50 

197.921 

4-367 

0.640138 

7 

231 

3  32  09 

230.837 

6.716 

0.827136 

8 

264 

4  02  28 

263.694 

9.784 

0.990505 

9 

297 

4  32  48 

296.465 

13.662 

1-135529 

JO 

33« 

5  03  07 

329.117 

18.445 

1.265884 

n 

363 

5  33  27 

361.607 

24.223 

1.384226 

12 

396 

6  03  47 

393.886 

31-084 

1.492536 

6  3.4  07 

c.  CHORD-LENGTH  =  34. 

n. 

nc. 

&* 

y. 

X. 

Log.  x. 

I 

34 

o°  29'  25" 

34.000 

0.0495 

8.694175 

2 

68 

o  58  49 

67.999 

.2473 

9093I43 

3 

102 

i  28  14 

101.996 

.6923 

9.840294 

4 

136 

i  57  39 

135.987 

1.483 

O.I7J27I 

5 

170 

2  27  04 

169.965 

2.719 

0.434481 

6 

204 

2  56  2() 

203.918 

4.499 

0.653103 

7 

233 

3  25  55 

237.832 

6  920 

0,840101 

8 

272 

3  55  20 

271.685 

10.080 

1.003470 

9 

306 

4  24  46 

305.449 

14.076 

1.148494 

10 

340 

4  54  12 

339.090 

19.004 

1.278849 

ii 

374 

•5  23  38 

372.565 

24-957 

1.397191 

12 

408 

5  53  05 

405.822 

32.026 

1.505501 

6  22  II 

70 


TABLE    III. 


c.  CHORD-LENGTH  =  35. 

;/. 

11C. 

Ds. 

y- 

A\ 

Log  a\ 

I 

35 

o°  28'  34" 

35.000 

0.0509 

8.706764 

2 

70 

o  57  09 

69.999 

•2545 

9.405732 

3 

105 

i  25  43 

104.996 

.7127 

9.852883 

4 

140 

I  54  17- 

139.987 

1.527 

0.183860 

5 

175 

2  22  52 

174.964 

2.799 

0.447070 

C 

210 

2  51  27 

209.916 

4.631 

0.665692 

7 

245 

3  20  01 

244.827 

7.123 

0.852690 

8 

280 

3  4S  36 

279.675 

10.377 

.016059 

9 

3J5 

4  17  12 

314-433 

14.490 

.161083 

10 

35° 

4  45  47 

349-  °63 

I9-563 

.291438 

ii 

385 

5  14  23 

383-523 

25.691 

.409780 

12 

420 

5  43  oo 

417.758 

32.968 

.518090 

6  09  36 

\ 

c.  CHORD-LENGTH  =  36. 

//.   tic. 

Ds. 

y- 

X. 

Log  jc. 

I 

36 

o°  27'  47" 

36.000 

O.O524 

8.718998 

2 

72 

o  55  33 

71.999 

.26l8 

9.417967 

3 

io3 

I  23  20 

107.996 

•7330 

9.865118 

4 

144 

I  51  07 

143.987 

I-57I 

0.196095 

5 

1  80 

2  18  54 

179.963 

2.879 

0.459304 

6 

216 

2  46  41 

215.913 

4-  764 

0.677927 

7 

252 

3  14  28 

251.822 

7.327 

0.864924 

8 

2:8 

3  42  15 

287.666 

10.673 

1.028293 

9 

324 

4  10  03 

323-417 

14.905 

1.173318 

10 

360 

4  37  5i 

359037 

20.  122 

1.303673 

ii 

396 

5  05  39 

394.480 

26.425 

1.422014 

5  33  27 

TABLE  III. 


11. 

11C, 

z>* 

/• 

x. 

Log  x. 

I 

37 

o°  27'  02" 

37.000 

0.0538 

8.730898 

2 

74 

o  54  03 

73-999 

.2691 

9.429866 

3 

in 

I  21  05 

110.996 

•7534 

9.877017 

4 

148 

I  48  07 

147.986 

1.614 

0.207994 

5 

185 

2  1  5  09 

184.962 

2-959 

0.471203 

6 

222 

2  42  II 

221.911 

4.896 

0.689826 

7 

259 

3  09  ]3 

258.817 

7-530 

0.876824 

8 

296 

3  36  15 

295.657 

10.970 

1.040193 

9 

333 

4  03  17 

332.400 

15-319 

1.185217 

10 

370 

4  30  20 

369.010 

20.681 

L3I5572 

ii 

407 

4  57  23 

405-438 

27.159 

I.4339I3 

5  24  26 

e.  CHORD-LENGTH  =  37. 


c.  CHORD-LENGTH  =  38. 


11. 

m\ 

D* 

* 

x» 

Log  jr. 

i 

38    o''  26'  19" 

38.000 

0-0553 

8.742480 

2 

76 

o  52  39 

75-999 

.2763 

9.441448 

3 

114 

i  18  57 

113.996 

•  7737 

9.888599 

4 

152 

i  45  16 

151.986 

1.658 

0.219576 

5 

190 

2  ii  35 

189.961 

3.039 

0.482785 

6 

228 

2  37  54 

227.909 

5.028 

0.701408 

7 

266 

3  04  14 

265.812 

7-734 

0.888406 

8 

304 

3  30  33 

303.648 

11.266 

1.051774 

9 

342 

3  56  53 

34L384 

15.733 

1.196799 

10 

380 

4  23  13 

378.983 

21.240 

I.327T54 

ii 

418 

4  49  33 

416.396 

27.893 

1-445495 

5  15  53 

72 


TABLE   III. 


c.     CHORD-LENGTH  =  39. 

//. 

nc. 

D*. 

>'• 

X. 

Log  x  . 

I 

39 

o°  25'  38" 

39.000 

0.0567 

8.753761 

2 

78 

o    51   17 

77-999 

.2836 

9.452729 

3 

117 

I    16   55 

116.996 

.7941 

9.899880 

4 

156 

I    42    34 

I55.985 

1.702 

0.230857 

5 

*95 

2     08     13 

194.960 

5.119 

0.494066 

6 

234 

2    33    51 

233.906 

5.160 

0.712689 

7 

273 

2    59    30 

272.807 

7.938 

0.899687 

8 

312 

3    25    09 

311.638 

11.563 

1.063055 

9 

35i 

3    50   48 

350.368 

16.147 

1.208080 

10 

390 

4    16   28 

388.956 

21.799 

L338435 

4    42   07 

c.     CHORD-LENGTH  ~  40. 

n. 

1IC. 

Ds. 

}'• 

X. 

Log  x. 

I 

40 

o°  25'  oo'' 

40.  ooo 

0.0582 

8.764756 

2 

80 

o    50  oo 

79-999 

.2909 

9.463724 

3 

120 

I    15   oo 

1  1  9.  996 

.8145 

9.910875 

4 

1  6O 

I    40  oo 

I59-985 

1.745 

0.241852 

5 

2OO 

2     05    CO 

199-959 

3.199 

0.505062 

6 

240 

2     30    01 

239.904 

5.293 

0.723684 

7 

280 

2    55  oi 

279.802 

8.141 

0.910682 

8 

32O 

3    20  oi 

319.629 

11.859 

1.074051 

9 

360 

3    45   02 

359-352 

16.561 

1.219075 

10 

4OO 

4    10  03 

398.929 

22.358 

1.349430 

4    35   03 

c.     CHORD-LENGTH  =  41. 

n. 

nc. 

/>* 

•  y- 

X. 

Log  x  . 

i 

41 

o°  24'  24" 

41.000 

0.0596 

8.775480 

2 

82 

o   48  47 

81.999 

.2982 

9.474448 

3 

123 

i    13   10 

122.996 

.8348 

9.921599 

4 

164 

i    37   34 

163  985 

1.789 

0.252576 

5 

205 

2    oi    57 

204.958 

3-*79 

0.515786 

6 

246 

2     26    21 

245.901 

5-425 

o.  734408 

7 

287 

2    50  45 

286.797 

8.345 

0.921406 

8 

328 

3    15  09 

327.620 

12.156 

1.084775 

9 

369 

3    39  33 

368.336 

16.975 

1.229799 

10 

410 

4    03   57 

408.903 

22.917 

1.360154 

4    28   21 

73 


TABLE    III. 


c.     CHORD-LENGTH  =  42. 

n. 

11  r. 

Ds. 

/• 

X. 

Log  x. 

i 

42 

o°  23'  49" 

42.000 

0.0611 

8.785945 

2 

84 

o   47   37 

83.999 

.3054 

9.484913 

3 

126 

i    n    26 

125.996 

.8552 

9.932065 

4 

168 

i    35    14 

167.984 

1.832 

0.263042 

5 

2IO 

i    59  02 

209.957 

3-359 

0.526251 

6 

252 

2     22    52 

251.899 

5-557 

0.744874 

7 

294 

2     46    41 

293.792 

8.548 

0.931871 

8 

336 

3-  10  30 

335.6ii 

12.452 

1.095240 

9 

373 

3    34   19 

377.319 

17-389 

1.240265 

10 

420 

3    58   08 

418.876 

23.476 

1.370619 

4    21    57 

f.     CHORD-LENGTH  =  43. 

«. 

nc. 

/>*. 

y> 

X. 

Log  x. 

i 

43 

o°  23'  15" 

43.000 

0.0625 

8.796164 

2 

86 

o   46  31 

85.999 

•3127 

9-495I33 

3 

129 

I   09  46 

128.996 

.8755 

9.942284 

4 

172 

i    33  02 

171.984 

1.876 

0.273261 

5 

215 

i    56   17 

214-955 

3-439 

0.536470 

6 

2^8 

2    19   33 

257-897 

5-690 

0.755093 

7 

301 

2     42    48 

300.787 

8-752 

0.942090 

8 

344 

3    06  04 

343.601 

12.749 

1.105459 

9 

3S7 

3    29   20 

386.303 

17.803 

1.250484 

10 

430 

3    52   35 

428.849 

24.035 

1.380839 

4    15    50 

c.     CHORD-LENGTH  =  44. 

n. 

nc. 

Ds. 

}'• 

X. 

Logx. 

i 

44 

o°  22'  44" 

44.000 

0.0640 

8.806149 

2 

88 

o    45   27 

87.999 

.3200 

9-505117 

3 

132 

i    08    ii 

131-995 

.8959 

9.952268 

4 

176 

i    30   55 

175.984 

1.920 

0.283245 

5 

220 

i    53   38 

219-954 

3-519 

0.546454 

6 

264 

2     l6    22 

263.894 

5.822 

0.765077 

7 

308 

2    39  06 

307.78, 

8.955 

0.952075 

8 

352 

3   oi    50^. 

351-592 

13.045 

1.115444 

9 

396 

3    24   34 

395.287 

18.217 

1.260468 

3    47    18 

74 


TABLE    III. 


c.     CHORD-LENGTH  =  45. 

ti. 

7l('. 

z£ 

y- 

jr. 

Log  jr. 

45 

o°  22'  13" 

45.000 

0.0655 

8.815908 

2 

9° 

o   44   27 

89.999 

.3272 

9-5I4877 

3 

135 

i    06  40 

134-995 

.9163 

9.962028 

4 

i  So 

i    28   53 

179.983 

1.963 

o.  293005 

5 

225 

i    5i   07 

224.953 

3-599 

0.556214 

6 

270 

2     13    20 

269.892 

5.954 

0.774837 

7 

315 

2    35   34 

314.778 

9-159 

0.961834 

8 

360 

2    57  43 

359-583 

I3.34I 

1.125203 

9 

405 

3    20  01 

404.271 

18.631 

1.270228 

3    42    15 

c.     CHORD-LENGTH  ±~  46. 

;/. 

nc. 

Ds. 

y- 

jr. 

Log  x. 

I 

46 

0°  2l'  44" 

46.  ooo 

0.0669 

8.825454 

2 

92 

o    43   29 

91.999 

•  3345 

9-524422 

3 

138 

i    05-13 

137.995 

.9366 

9.971573 

4 

184 

i    26   58 

183.983 

2.007 

0.302550 

5 

230 

i    48  42 

229.952 

3.679 

0.565759 

6 

276 

2     10    26 

275.889 

6.087 

0.784382 

7 

322 

2     32     II 

321.773 

9.362 

0.971380 

8 

368 

2    53   56 

367.573 

13.638 

I.I34749 

9 

414 

3    15   40 

413.255 

I9-045 

1.279773 

3    37   24 

r.     CHORD-LENGTH  =  47. 

«. 

nc. 

Ds.    ' 

y- 

jr. 

Log  x. 

i 

47 

o°  21'  16" 

47.000 

0.0684 

8.834794 

2 

94 

o   42   33 

93.999 

.3418 

9.533762 

3 

141 

i    03   50 

140.995 

•9570 

9.980913 

4 

188 

i    25   06 

187.982 

2.051 

0.311890 

5 

235 

i    46  23 

234.951 

3-759 

0.575100 

6 

282 

2     O7    40 

281.887 

6.219 

0.793722 

7 

329 

2    28    57 

•  28.768 

9.566 

0.980720 

8 

376 

2     50    14 

.375^564 

13-934 

1.144089 

9 

423 

3    ii   31 

422.238 

-9-459 

1.289113 

3    32  48 

75 


TABLE    III. 


f.     CHORD-LENGTH  -  48. 

n. 

nc. 

zv 

'?• 

jr. 

Log^r. 

i 

48 

o°  20'  50" 

48.000 

o.  0698 

8.843937 

2 

96 

o   41   40 

95-999 

•  3491 

9-  542905 

3 

144 

I     02    30 

143.995 

•9774 

9.990057 

4 

192 

I     23    20 

191.982 

2.094 

0.321034 

5 

240 

i    44   10 

239-950 

3.839 

0.584243 

6 

288 

2     05    OO 

287.885 

6.351 

0.802866 

7 

336 

2     25    51 

335-7^3 

9.769 

0.989863 

8 

3S4 

2     46    41 

3S3.555 

14.231 

1.153232 

3    06  31 

f.     CHORD-LENGTH  =  49. 

ft. 

nc. 

Ds. 

y- 

jr. 

Log  x. 

I 

49 

o°  20'  25" 

49.000 

0.0713 

8.852892 

2 

q8 

o   40  49 

97.999 

.3563 

9.551860 

3 

147 

i    01    14 

146.995 

.9977 

9.999011 

4 

196 

I     21     38 

195.982 

2.138 

0.329988 

5 

245 

I     42    03 

244.949 

3.919 

0.593198 

6 

294 

2     02    27 

293.882 

6.484 

0.811820 

7 

343 

2     22     52 

342.758 

9-973 

0.998818 

8 

392 

2    43    17 

39!-546 

14.527 

1.162187 

3   03   31 

c.     CHORD-LENGTH  =  50. 

n. 

nc. 

/?* 

y- 

X. 

Log  x. 

I 

50 

o°  20'  oo" 

50.000 

0.0727 

8.861666 

2 

100 

o   40  oo 

99-999 

.3636 

9.  560634 

3 

150 

I     OO    OO 

149.995 

1.018 

0.007785 

4 

200' 

I     20    00 

199.981 

2.182 

0.338762 

5 

250 

i    40  oo 

249.948 

3-999 

0.601972 

6 

3OO 

2     00    00 

299.880 

6.616 

0.820594 

7 

35° 

2     2O    OO 

349-753 

10.176 

1.007592 

8 

400 

2     40    OO 

399.536 

14.824 

1.170961 

3   oo  oo 

76 


TABLE   IV. 


FUNCTIONS  OF  THE  ANGLE  s. 


n. 

S. 

cos  s. 

log  vers  s. 

R  i°  x 

vers  s. 

sin  s. 

log  sin  s. 

s. 

I 

o°  10' 

•99999 

4.626422 

.024 

.00291 

7.463726 

o°  10' 

2 

o  30 

.99996 

5.580662 

.218 

.00873 

7.940842 

o  30 

/? 

I  00 

.99985 

6.182714 

•873 

-01745 

8.241855 

I  00 

4 

I  40 

.99958 

6.626392 

2.424 

.  02908 

8.463665 

I  40 

5 

2  30 

.99905 

6.978536 

5-453 

.  04362 

8.639680 

2  30 

6 

3  30 

.99813 

7.720726 

10.687 

.06105 

8.785675 

3  30 

7 

4  40 

.99668 

7.520498 

18.994 

.08136 

8.910404 

4  40 

8 

6  oo 

.99452 

7.738630 

31  383 

•10453 

9.019235 

6  oo 

9 

7  30 

.99144 

7.932227 

49.018 

•13053 

9.115698 

7  30 

10 

9  10 

.98723 

8.106221 

73-173 

.15931 

9.202234 

9  10 

ii 

II  00 

.98163 

8.264176 

105.270 

.19081 

9.280599 

II  00 

12 

13  oo 

•97437 

8  408748 

146.857 

.22495 

9.352088 

13  oo 

13 

15  10 

.9651718.541968 

199.570 

.26163 

9.417684 

15  10 

14 

17  30 

.953728.665422 

265.186 

.30071 

9.478142 

17  30 

15 

20  oo 

.93969 

8.780370 

345-540 

.  34202 

9-534052 

20  00 

16 

22  40 

.92276 

8.887829 

442-543 

•38537 

Q-585877 

22  40 

17 

25  30 

•90259 

8.988625 

558.153 

•43051 

9.633984 

25  30 

18 

28  30 

.87882 

9.083441 

694-335 

.47716 

9.678663 

28  30 

19 

31  40 

.85112 

9.172846 

853-0501 

.52498:  9.720140 

31  40 

20 

35  oo 

.81915 

9-2573*4 

1036.20  1 

-57358 

9-758591 

35  oo 

77 


TABLE 


SELECTED   SPIRALS   FOR   A   2°   CURVE,    GIVING 

A 

s. 

n  x  c. 

-As(w+l). 

£>'. 

flC 

10° 

1°  00' 

3  x  32 

2°  05'  OO" 

2°  03' 

41.12 

10 

I    40 

4  x  39 

2     08    13 

2     09 

61.04 

10 

2     3O 

5  x  43 

2    19  33 

2     IS 

73.69 

10 

3    30 

6  x  45 

2    35  34 

2     33 

78.81 

10 

4   40 

7  x  44 

3    oi  50 

2     40 

70.47 

20 

I     OO 

3  x  33 

2     01    13 

2     OI 

45.28 

20 

I    40 

4  x  41 

2    oi  57 

2     02 

73.85 

20 

2     30 

5  x  48 

2     05    00 

2     05 

99.99 

20 

3    30 

6  x  50 

2     20  00 

2     06 

109.52 

30 

I     OO 

3  x  34 

i    57  39 

2     OI 

46.14 

30 

I    40 

4  x  41 

2   oi  57 

2     01 

75.16 

30 

2     30 

5  x  49 

2     02    27 

2     02 

109.78 

30 

3    30 

6  x  50 

2     2O   OO 

2     02 

115.63 

30 

3    30 

6  x  50 

2     20   OO 

2     03 

110.90 

40 

I     00 

3  x  35 

I     54    17 

2     01 

46.90 

40 

I    40 

4  x  42 

i    59  02 

2     01 

76.96 

40 

2     30 

.5  x  50 

2     00   00 

2     01 

117.87 

78 


V. 


EQUAL   LENGTHS   BY   CHORD    MEASUREMENT. 

•£  old  line. 

^  new  line. 

Diff. 

X. 

h. 

£ 

291.12 

291.12 

.00 

.6516 

.040 

.061 

311-04 

311-04 

.00 

1.702 

.187 

.110 

323.69 

323.70 

+   .01 

3.439 

•354 

.103 

328.81 

328.82 

+   .01 

5.954 

•  59° 

.099 

320.47 

320.50 

-h  .03 

8.955 

.897 

.100 

545.28 

545.28 

.00 

.6719 

.122 

.182 

573-85 

573.84 

—   .01 

1.789 

.118 

.066 

599-99 

600.  oo 

+   .01 

3.839 

.527 

.137 

609.52 

609.52 

.00 

6.616 

•554 

.084 

796.14 

796.22 

+  .08 

.6923 

.566 

.082 

825.16 

825.16 

.00 

1.789 

.227 

.127 

859.78 

859'75 

-  .03 

3.919 

•  377 

.096 

865.63 

865.57 

-  .06 

6.616 

.249 

.038 

860.90 

860.98 

+  .08 

6.616 

1.013 

.153 

1046.90 

1047.15 

+  .25 

.7127 

1.222 

1.715 

1076.96 

1077.09 

+  .13 

1.832 

.848 

.463 

1117.87 

1117.77 

—   .10 

3.999 

.141 

.035 

79 


TABLE 


SELECTED  SPIRALS  FOR  A  4°  CURVE,  GIVING 

A 

S. 

72  X  C. 

-£>«(n  +  l). 

D\ 

d. 

10° 

i°  oo' 

3  x  16 

4°  10'  03" 

4°  07' 

20.22 

10 

I  40 

4  -  19 

4  23  13 

4  16 

29.12 

10 

2   30 

5  X  22 

4  32  48 

4  39 

38.75 

10 

3  30 

6  x  23 

5  04  26 

5  17 

41.37 

20 

I  40 

4  x  20 

4  10  03 

4  04 

34.92 

2O 

2   30 

5  x  24 

4  10  03 

4  09 

50.72 

20 

3  30 

6  x  27 

4  !9  19 

4  H 

63-69 

20 

4  40 

7  x  30 

4  26  44 

4  3i 

78.07 

20 

6  oo 

8  x  31 

4  50  24 

4  46 

81.88 

2O 

7  30 

9  x  32 

5  12  36 

5  16 

85.40 

30 

i  40 

4  x  20 

4  I0  03 

4  02 

35-57 

30 

2   30 

5  x  25 

4  oo  03 

4  04 

57-39 

30 

3  30 

6  x  28 

4  10  03 

4  07 

72.37 

30 

4  40 

7  x  32 

4  10  03 

4  14 

93-09 

30 

6  oo 

8  x  35 

4  17  12 

4  23 

110.31 

30 

7  30 

9  x  37 

4  30  20 

4  34 

122.  2O 

30 

9  10 

10  x  38 

4  49  33 

4  47 

126.86 

40 

2   30 

5  x  25 

4  oo  03 

4  02 

58.91 

40 

3  30 

6  x  28 

4  10  03 

4  04 

73-75 

40 

4  40 

7  x  32 

4  10  03 

4  08 

94.65 

40 

6   GO 

8  x  36 

4  10  03 

4  12 

121.33 

40 

7  30 

9  x  39 

4  16  28 

4  17 

142.86 

40 

9  10 

10  x  41 

4  28  21 

4  26 

154.34 

60 

2  30 

5  x  25 

4  oo  03 

4  01 

59-68 

60 

3  30 

6  x  29 

4  01  26 

4  02 

81.04 

60 

4  40 

7  x  32 

4  10  03 

4  03 

99-59 

60 

6  oo 

8  x  36 

4  10  03 

4  05 

125.81 

60 

7  30 

9  x  40 

4  10  03 

4  08 

154-42 

80 

2   30 

5  x  25 

4  oo  03 

4  or 

58.29 

80 

3  30 

6  x  29 

4  01  26 

4  01 

82.82 

80 

4  40 

7  x  33 

4  02  28 

4  02 

106.99 

80 

6  oo 

8  x  37 

4  03  17 

4  03 

I35.6I 

80 

7  30 

9  x  41 

4  03  57 

4  05 

164.79 

80 


V. 


f                   fuinvEESiTY- 

Vv  £*,d         o:H'         *t  ^ 
EQUAL    LENGTHS    BY   cfeQjto  ^JTASUR^kENT. 

•£  old  line. 

|  new  line. 

Diff. 

X. 

h. 

k. 

145-22 

145.17 

-  -05 

.3258 

.045 

•135 

154-12 

I54-I3 

+    .01 

-.8290 

.080 

.100 

163.75 

163.76 

+    .01 

1.760 

.177 

.100 

166.37 

166.39 

+  .02 

3.043 

•  305 

.100 

284.92 

284.92 

.00 

.8726 

.081 

.100 

300.  72 

300.72 

.00 

1.920 

.184 

.096 

313.69 

313.75 

+  .06 

3-573 

•375 

.iQ5> 

328.07 

328.08 

+    .01 

6.106 

.598 

.098 

332.88 

33L92 

+  .04 

9.191 

.910 

.092 

335-40 

335-47 

+  .07 

13-248 

1.310 

.099 

410.57 

410.57 

.00 

.8726 

•137 

.157 

432-39 

432.38 

—  .01 

2.000 

.147 

.074 

447-37 

447-35 

—    .02 

3.705 

.284 

.077 

468.09 

468.09 

.00 

6.513 

.687 

.105 

4S5.3I 

485-32 

s+  .01 

10-377 

1.091 

.105 

497.20 

497.23 

+  .03 

15.319 

1.526 

.100 

501.86 

5QJ.95 

H-  .09 

2T.240 

2.126 

.100 

558.91 

558.88 

—  .03 

2.000 

.109 

.054 

573-75 

573-74 

—   .01 

3-705 

.361 

.097 

594.65 

594-66 

+    .01 

6.513 

•977 

.150 

621.38 

621.33 

—  .05 

10.673 

•973 

.091 

642.86 

642.83 

-  .03 

16.147 

1.  100 

.086 

654.34 

654-36 

+    .02 

22.917 

2.186 

.095 

809.68 

809.67 

—    .01 

2.000 

.180 

.090 

831.04 

831.03 

—   .01 

3.837 

.461 

.120 

849.59 

849-52 

—  .07 

6.513 

•572 

.088 

875.81 

875-76 

-  .05 

10.673 

1.074 

.106 

904.42 

904.36 

-  .06 

16  561 

1.718 

.104 

1058.29 

1058.61 

-f-  .32 

2.OOO 

•979 

.490 

1082.82 

1082.71 

—  .11 

3.837 

.295 

.074 

1106.99 

1107.03 

+  .04 

6.716 

1.  000 

.149 

1135-61 

H35.5I 

—  .10 

10.970 

1.199 

.109 

1164.79 

1164.92 

+  .13 

16.975 

2.440 

.144 

81 


TABLE 


SELECTED  SPIRALS  FOR  AN  8°  CURVE,  GIVING 

A 

S. 

n  X  c. 

•#fi(7l  +  l). 

D'. 

d. 

10° 

2°  30' 

5x11 

9°  06'  01" 

9°  06' 

19-95 

20 

2   30 

5  x  12 

8  20  26 

8  16 

25.71 

2O 

3   30 

6  x  14 

8  20  26 

8  34 

34-86 

20 

4  40    7  x  15 

8  53  5i 

8  54 

39-90 

20 

6  oo 

8  x  16 

9  23  07 

9  24 

45-52 

30 

2   30 

5  x  12 

8  20  26 

8  07 

26.50 

30 

3  30 

6  x  14 

8  20  26 

8  14 

36.16 

30 

4  40 

7  x  16 

8  20  26 

8  26 

47.01 

30 

6  oo 

8  x  17 

8  49  55 

8  36 

53-13 

30 

7  30 

9  x  18 

9  16  08 

8  46 

60.05 

30 

9  10 

10  x  19 

9  39  36 

9  14 

65.70 

40 

2   30 

5  x  12 

8  20  26 

8  04 

26.93 

40 

3  30 

6  x  14    8  20  26 

8  08 

36.85 

40 

4  40 

7  x  16 

8  20  26 

8  14 

48.25 

40 

6  oo 

8  x  18 

8  20  26 

8  22 

61.35 

40 

7  30 

9  x  19 

8  46  49 

8  30 

68.07 

40 

9  10 

10  X  20 

9  10  34 

8  40 

75-01 

40 

II   00 

II  X  21 

9  32  03 

8  54 

82.13 

40 

13  oo 

12  X  22 

9  5i  36 

9  H 

89.81 

60 

2   30 

5  x  12 

8  20  26 

8  02 

27.30 

60 

3  30 

6  x  14 

8  20  26 

8  03 

38.22 

60 

4  40 

7  x  16 

8  20  26 

8  06 

49-75 

60 

6  oo 

8  x  18 

8  20  26 

8  10 

62.87 

60 

7  30 

9  x  20 

8  20  26 

8  16 

77.16 

-  60 

9  10 

10  X  22 

8  20  25 

8  24 

93.05 

60 

II   00 

II  X  23 

8  42  13 

8  31 

101.08 

60 

13  °° 

12  X  25 

8  40  28 

8  48 

118.19 

60 

15  10 

13  x  26 

8  58  59 

9  02 

127.21 

60 

17  30 

14  x  27 

9  16  07 

9  22 

136.45 

80 

4  40 

7  x  17 

7  50  57 

8  04 

57.04 

80 

6  oo 

8  x  19 

7  54  03 

8  06 

71.78 

80 

7  30 

9  x  20 

8  20  26 

8  oSJ- 

79.18 

80 

9  10 

IO  X  22 

8  20  25 

8  13 

95.23 

80 

II   OO 

ii  x  24 

8  20  25 

8  19 

112.67 

80 

13  oo 

12  X  26 

8  20  25  . 

8  28 

130.86 

80 

15  10 

13  x  27 

8  38  59 

8  34 

140.88 

80 

17  30 

14  x  28 

8  56  13 

8  42 

150.55 

82 


EQUAL    LENGTHS   BY    CHORD   MEASUREMENT. 

%  old  line. 

£  new  line. 

Diff. 

X. 

h. 

k. 

82.45 

82.47 

+    .02 

.8798 

.051 

.058 

150.71 

150.72 

+    .01 

-9598 

.051 

.053 

159.86 

I5Q-88 

+    .02 

1.852 

.117 

.063 

164.90 

164.92 

+    .02 

3-C53 

.185 

.061 

170.52 

170.55 

-f     .03 

4-744 

.221 

.047 

214.00 

214.00 

.CO 

.9598 

.049 

.051 

223.66 

223.68 

+    .02 

1.852 

.142 

.077 

234.51 

234.53 

+    .02 

3-256 

.260 

.080 

240.  63 

240.65 

+    .02 

5.040 

.325 

.065 

247-55 

247-55 

.00 

7-452 

.287- 

•039 

253.20 

253.18 

—  .02 

10.620 

•590 

.056 

276.93 

276.94 

+    .01 

.9598 

.079 

.082 

286.85 

286.87 

+    .02 

1.852 

.181 

.098 

298.25 

298.24 

—   .01 

3.256 

•293 

.090 

3H-35 

3X1-33 

—   .02 

5.337 

.330 

.062 

318.07 

318.06 

—   .01 

7.866 

.472- 

.c6o 

325-01 

325.00 

•v  —   .OI 

11.179 

.629 

.056 

332.13 

332.12 

—   .01 

15.415 

.840 

•054 

339-81  ' 

339-81 

.00 

20.723 

I.O24 

.049 

402.  30 

402.32 

+    .02 

.9598 

.136 

.142 

413.22 

413.19 

-   .03 

1.852 

.083 

•045 

424.75 

424.76 

+   .01 

3-256 

.317 

.097 

437.87 

437.88 

+   .01 

5-337 

•539 

.101 

452.16 

452.18 

+   .02 

8.280 

.863  - 

.104 

468.05 

468.02 

-  .03 

12.297 

I-I39 

.093 

476.08 

476.09 

+    .01 

16.883 

1.523 

.090 

493-19 

493.18 

—   .01 

23.548 

2.160 

.092 

502.21 

502.21 

.00 

30.817 

2.613 

.085 

5H-45 

5H.45 

.00 

39-595 

3-157 

.080 

557.04 

557.02 

—  .02 

3.460 

.366 

.106 

57L78 

57L75 

-  .03 

5*633 

.408 

.072 

579^8 

579-  18 

.00 

8.280 

.860 

.104 

595.23 

595.25 

H-  .02 

12.297 

1.346 

.110 

612.67 

612.70 

+  .03 

17.617 

1.719  - 

.109 

630.86 

630.90  » 

+  .04 

,24.490 

2.738 

.112 

640.  88 

640.88 

.00 

32.002 

3.H9 

.098 

650.55 

650.62 

+  .07 

41.062 

3.809 

.093 

TABLE 


SELECTED  SPIRALS  FOR  A  16°  CURVE, 

A 

S. 

n  X  c. 

.£>s(n  +  l). 

D\ 

d. 

30° 

4°  4o' 

7  x  10 

13°  2l'  48" 

18°  oo' 

33-59 

40 

6  oo 

8  x  10 

15  02  34 

17  14 

36.14 

60 

7  3o 

9  x  10 

16  43  3i 

16  32 

38.47 

60 

9  10 

10  X  11 

16  43  3i 

16  48 

46.40 

60 

II   00 

II  X  12 

16  43  3i 

17  14 

54-62 

60 

13  oo 

12  X  12 

18  07  48 

17  22 

54-  M 

60 

15  10 

13  x  13 

18  oi  18 

18  10 

62.88 

60 

17  30 

14  x  13 

19  19  14 

18  12 

62.85 

60 

20  oo 

15  x  14 

19  06  05 

20  00 

72.14 

80 

7  30 

9  x  10 

16  43  31 

16  16 

39-74 

80 

9  10 

IO  X  II 

16  43  31 

16  26 

47-49 

80 

II   OO 

II  X  12 

16  43  31 

16  38 

56.19 

80 

13  oo 

12  X  13 

16  43  30 

16  56 

65.24 

80 

15  10 

13  x  14 

16  43  29 

17  22 

74.72 

80 

17  30 

14  x  14 

17  55  44 

17  24 

75.02 

80 

20   00 

15  x  15 

J7  50  54 

18  06 

85.15 

So" 

22   40 

16  x  15 

18  58  25 

18  08 

85-18 

80 

28   30 

18  x  16 

19  53  20 

19  42 

95.84 

84 


GIVING  EQUAL    LENGTHS   OF    ACTUAL   ARCS. 

•£  old  line. 

£  new  line. 

Biff. 

X. 

h. 

k. 

127.64 

127.64 

.00 

2.035 

.388 

.191 

I6I.55 

161.55 

.00 

2.965 

.430 

.145 

226.58 

226.56 

—    .02 

4.140 

.436 

.105 

234.50 

234.45 

-  .05 

6.148 

.576 

.094 

242.73 

246.67 

—  .06 

8.808 

.860 

.099 

242.25 

242.26 

+    .01 

11-303 

1.093 

.097 

250.99 

250.99 

\       .00 

15.409 

1.516 

.098 

250.96 

250.97 

+    .01 

19.064 

1-552 

.081 

260.25 

260.25 

.00 

25-031 

2.182 

.087 

290.55 

290.47 

-  .08 

4.140 

,328 

•305 

298.30 

298.27 

-  -03 

6.148 

.680     1     .Hi 

307.01 

306.96 

-  .05 

8.808 

-943 

.107 

316.06 

316.03 

—  .03 

12.245 

1.384 

•  113 

325.53 

325-54 

-f    .01 

16.594 

1-973 

.119 

325.83 

325-81 

—  .02 

20.531 

1-939 

,094 

335-97 

335-96 

—   .01 

26.819 

2.657 

.099 

336.00 

335-99 

—  .01 

32.276 

2.677 

.083 

346.65 

346.66 

+  .01 

48.221 

3.748 

.078 

1 

THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 

AN  INITIAL  PINlToF  25  CENTS 

^!,L«BBE  ASSESSED  F0*   FAILURE  TO   RETURN 
THIS  BOOK  ON   THE  DATE  DUE.   THE  PENALTY 

DAV'-A^RETASE  T0  50  CENTS  ON  ™«  ™ 

OVERDUE^.  "•°°    °N     THE    8EVEN™ 


I8Apr52l?J 


JUL  27    1943 


Yft 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


